HUGOC9999

I am not sure it is as high 50/50, but yes in theory a CO can demand your original evidence of U.S. citizenship.  
 
In your case this would either be a certificate of citizenship or the evidence that was used to get your first U.S. passport.  
 
 The  department of state issued you your first passport and is the same entity that is going to issue your spouse’s visa. 
 
Thus I think it is more likely the state department will review your file to re-verify your status than it is that it will issue an RFE.  

You don't get the naturalization certificate because you did not naturalize but became the citizen thru father. Passport is enough to prove US citizenship. However if you want to have something as a proof that never expires, you can apply for certificate of citizenship thru N600 but not necessary.

Greetings!
Just make sure that it is still valid and not expired. NG

As long as you already have a valid US passport, that will be your proof of citizenship without any issues. It should be accepted just about anywhere as proof. I would save the money, and not worry about the actual certificate.

Good answer.  I want to address what I made bold in the OP's original post. 
 
From what I'm aware a joint sponsor is someone who doesn't live in the same household you intend to live in with your spouse, you don't necessarily have to live with them once you arrive in the US, and it could even be someone from a state other than the one you plan to live in. Am I correct?
 
 
No and yes.  NO, a joint sponsor CAN live in the same house as you, but you are NOT combining income with them.  Living in the same house does not necessarily make a person a household member.  Also actually qualifying as a household member does not usually mean they should act as household member instead of a joint sponsor. 

YES, they can be any US legal resident or citizen living anywhere in the US, and you don't need to plan to live with them.
 
Not surprised your potential joint sponsor is having doubts.  Maybe they have seen and read the actual form/contract they will sign???
 
In reality, if the joint sponsor trusts you and your spouse to support yourselves and is confident you can and will, there is little risk.  If not, acting as joint sponsor would be quite foolish.

The joint sponsor has to submit everything that you, as the primary sponsor, submit.  This includes the I-864 with all the supporting wage/tax documentation, etc.  It is pretty well spelled out in the instructions.
 
The joint sponsor would be signing a contract with the US government that if the sponsored immigrant were to use any means-tested benefits they are not eligible for, sponsor is on the hook for that money.  So, it is kind of a big deal, and not some casual "signature only" situation.
 
If the joint sponsor is not a close family member, or very close friend, and does not know the immigrant personally, expect scrutiny at the consular level.
 
If it were me, I would do everything in my power to increase my own income before asking anyone other than my mom, for example.

I'm pretty sure you both, petitioner and joint sponsor, have to file I864.

The wife's birth certificate I believe is need at the NVC stage, not the i-130 filing with USCIS.
At the i-130 stage, you are proving you are a US Citizen or Resident, and your relationship with your spouse.
Also if you have been married before, you also need to include documentation showing that you were free to marry your spouse.

If you haven't already started the process, I would say do the CR1. My immigration attorney asked me if I wanted to do the CR1 over the K1 but he said it's not a good idea to switch from one to the other after you have already started. I went with the K1 because travel was restricted and my 2 years since my last seeing her in person was rapidly approaching. I am glad went with the K1 because travel to the Philippines is still a no go.

You read my mind, I've thought about actually going for the DDS and it's a long and gruesome process much like de CR1. Thanks for the help.

If you marry immediately, there is a chance it will take 2 years for everything to pan out.  Your wife would then enter the US as an Immediate Relative and receive a 10 year green card and be eligible to apply for citizenship after 3 years in country.  Even if she comes on a CR (conditional relative), she's able to work immediately.  If you choose the K-1 route, she'll enter the country and not be able to work for around 8 months.  It can put a lot of strain on a relationship and I don't know what it looks like for you to get your DDS licensure stated in the US.  She may be the bridging income if you have to write a board exam.  Just something to keep in mind. 

As others have said, the CR1 is the way you should go.  As for establishing domicile, Ive written on other threads on my experience on that.  I was out of the US for around 28 years and when I filed for the K1, I included a brief note that I was going to be moving back to the US between such and such months.  We never received any questions on that, however I did follow through with my plan that I gave to USCIS.  I only had retirement and brokerage accounts in the US as well.  No US bank accounts, no driver license, no US credit cards, no address, etc etc.  We never received any questions on this and I dont believe its something you should worry too much about.  
 
I did file taxes all of those years, however one thing you may want to check is if the IRS is logging in your overseas address correctly.  When I eventually returned to the US, they had a hard time establishing my identity as one of security questions I was being asked was my address.  This was at a time when the offices were closed due to Covid so I had to do this over the phone and it took about 3-4 hours and being transferred to multiple people.  

I lived in Ecuador the past 10 years and my fiance is Ecuadorian. We have gone the K1 route and are at 8 months waiting for approval of the first part (NOA2). We chose K1 as we did want to marry in the US and hoped it would be quicker (doubtful). Both are about the same right now time wise. CR1 seems to be slightly longer with the NVC stage, but it does have work advantages once in the US. I moved back to the US a few months ago to get resettled in preparation for him to be here and to make sure my domicile case was strong, but we know the wait for us is still going to be about 4-6 months if not more. We probably still would have gone with K1 vs CR1 as we want to be married in the US with my family and had just gotten engaged when we applied. We agreed to do "engaged time" separated as opposed to "married time" so it would be an easier transition for our children. Time wise, both options are about the same right now, but since we agreed that once we got married we wanted to always go be able to return home to the same place each day and live a normal life, we ultimately decided to do the K1. It has been a strain being apart, but I was still in Ecuador for the first 5 months of the wait so we would only be apart for about 6-8 months out of the 12 month plus process.  
 
With the deteriorating economic situation in Ecuador and my school year over half over when we filed (I was a teacher), we both decided it was better for me to come back a bit sooner and wait out the last part. 
 
Additionally, while I have permanent residency in Ecuador through my son, since I am a US citizen and COVID was in full swing, we would have had to wait much longer to file while I gathered all the paperwork for the registro civil in order to get married since the embassy stopped giving out the "single" declaration and I would have had to have gotten it from my local US county (there were no guarantees for us at the time that it wouldn't take many more months before we could get married as it was pre-mass vaccination), and we wanted to get our "place in line" as quickly as possible when we filed.  This part sounds like it wouldn't be an issue for you and it would be much easier and faster to get married and get your application in. 
 
Good luck with whatever you choose! My only advice would be to get paperwork in ASAP so it can all get started! 

Oh I meant convincing them of intent to re-establish domicile might be harder because you've lived so long outside the US (10 years I gather?), you probably don't have bank accounts or a mailing address that you've maintained? If you do have those, it will be a lot easier. Worst case scenario, you simply move back about 1-2 months before and establish that (address, bank account, etc) and it should be enough.
 
Don't worry about filing online, it's not related to domicile. Domicile comes way, way later when you're prepping stuff for the Embassy/NVC. You don't need to send anything in about domicile with the initial petition. You can file online no problem and it won't affect that.
 
Check out the guides to get a better idea of the steps involved:
https://www.visajourney.com/guides/
 

I did, in both countries. 

A US citizen is required to file US tax returns / no tax treaty with the US for Ecuador
u also need a US income to sponsor your fiancee or wife
and i would guess a joint sponsor since u can not show US income
Definitely I choose CR1 over K1
The advantages of the spouse visa far outway the fiancee
 
 
 
This is post from Crazycat
Every couple has their own priorities, and each couple must decide which visa is better for their situation.
K-1      
    More expensive than CR-1    
    Requires Adjustment of Status after marriage (expensive and requires a lot of paperwork)    
    Spouse can not leave the US until she/he receives approved Advance Parole (approx 5-6 months)    
    Spouse can not work until she/he receives EAD (approx 5-6 months)    
    Some people have had problems with driver licenses, Social Security cards, leases, bank account during this period    
    Spouse will not receive Green Card for many months after Adjustment of Status is filed.
    A denied K-1 is sent back to USCIS to expire
  
CR-1   
    Less expensive than K-1    
    No Adjustment of Status(I-485, I-131, I-765, I-944) required.    
    Spouse can immediately travel outside the US    
    Spouse is authorized to work immediately upon arrival.    
    Spouse receives Social Security Card and Green Card within 2 or 3 weeks after entering the US    
    Opening a bank account, getting a driver's license, etc. are very easily accomplished with GC, SS card, and passport.
    Spouse has legal permanent Resident status IMMEDIATELY upon entry to US.

Oh sorry, I just saw that somebody else had posted it. My bad. 
I don't have first hand experience with either the K1 or the CR1, just from what I read here. However, the CR1seems to be a better option right now because you can both wait it out by living in Ecuador. I don't think that's a possibility with a K1. And with a child in the middle, like you said, you really don't want to spend too much time apart.