No, you cannot work with your K-1 visa at all. Your Social Security card does not give authorization to work.
After you get married, you will apply for Adjustment of Status (AOS) and as a part of that package you will apply for a temporary work permission, called EAD (Employment Authorization Document). The EAD takes around 3 months, maybe less, and once you get your GC, you don't need the EAD anymore. The GC gives you authorization to work.
You may have read that K-1 is work authorized, and it is not. Technically, you may apply for work authorization (an EAD) based on your K-1 visa, but it will expire when your I-94 does. Since your I-94 will be valid for 3 months, and the EAD takes about 3 months to process, you might get your K-1 EAD a few days before it expires. In other words, it's basically useless.
So enter on K-1, get SSN, get married, apply for AOS with an EAD, and once you get that EAD you can work. The EAD you get based on pending AOS is good for a year, and you will get your GC before then. Don't plan on working before marriage at all, and for about 3 months after you apply for AOS.
Good luck.
