Southwest.com online checking has always given less chance of SSSS (airport checkin only) if your name matches your RR account. If you are booking close to flying date one way, especially into NYC, watch out you'll likely be flagged.

Southwest.com also gave (and is giving) a free name change update to your rapid rewards account in case it didn't match your government photo ID. This would in theory lower the risk for SSSS mismatch. But it may not take effect immediately, maybe just for new bookings

Maybe I dodged a bullet though on the other hand. I had our DMV change my government photo ID to mach my rapid reward acccount. I found it easier to do that since I needed a new photo card anyway.