Permanent wife

No.

it's my understanding
that an LPR can file the big adjustment of status case in the USA,
but
the beneficiary must remain in status on the visa that first got him through a port of entry.
AND
there was a window where F2A-types were 'current' on priority date last year, so if you attempted this last year when he was in status, it might have been possible, then.
Now? Not so much.
Since he's out of status and you are a LPR, you cannot file the big AOS set after marriage and him remain here during the processing whilst the overstay is forgiven when the greencard is issued - that's a priviledge reserved for USCitizens and their foreign spouses.
Anyway - that's the jist of it. An immigration attorney will tell you something similar, many consult for free, so try to find one soonish , interview many, as I think that ALTHOUGH there is no hope for processing for a greencard inside the USA for you, an immigration attorney can tell you the same thing.

An LPR cannot file for a fiancé. Furthermore, a K-1 would not help since the beneficiary is already in the US.
Marriage and filing the I-130 is a good first step. However, the beneficiary can not adjust with a PD that is not current.
Since the new fiscal year, the PD for the F2a category has not changes from Sept. 8. 2013. That's 6 months stuck on the same date.
Go get an immigration lawyer. This is not a DIY.

1. Say you marry him tomorrow. Since you are an LPR there are no benefits he can receive from you right away. He has to wait for the priority date on the spousal visa petition to become current then he has to interview for the visa in another country. Since he is out of status now even when the PD becomes current he cannot adjust from within the US.
2. If he is deported there will also be the need for a waiver or two depending on the length of his overstay.
3. Depending on the crime that may create problems also.
Best thing to do, marry now and start the paperwork for the I-130.