On the I-130, in Part B (Information about You) question 1 it ask for the current legal name. Have your partner leave the middle name field blank. (I know from experience since my wife doesn't have a middle name and always left it blank with no issue.) Then on question 7 when it ask for former names used, have your partner put down their name with the middle name (for example: First Middle Last).
USCIS will go off of the proof provided to show US Citizenship which would be the US issued birth certificate and/or the US Passport. If you put down a middle name and none is on the citizenship proof provided then (if they catch it) they may ask for proof though a court ordered name change. As a USC, I can't recall being asked by USCIS to provide a copy of my SS card, but they did ask for a copy of my US Passport and US Birth Certificate.
Does your partner have a drivers license or state issued ID card? Some put down the persons full name including their middle name. Does your partner have their "middle" name on their drivers license. I didn't want my middle name on my drivers license when I moved to Maryland, it wasn't on my NY driver license, and the clerk told they need to put down my full legal name, and if I don't want it on there then I need to get a name change with the courts.
In theory the SS card is incorrect, and your partner should file a new SS-5 with the social security office in order to get a corrected social security card without the middle name.
