Personally I wouldn't have liked the way you handled it. Mostly because the way you write implies that you never took her discomfort seriously. From the start to the end you thought her request was her being a sore loser, and you likely communicated that to your family.

Did you ask your girlfriend how she wanted to handle this? Or have any conversations with her about her feelings on the matter outside of a fighting situation?

Personally if I was her I wouldn't have wanted you to tell your family that the shit talking upset me, but for you as my boyfriend to not contribute and to 'have my back' in a way. Like come behind me and help me make shots and other cutesy shit that will draw away the sit talking for a bit. Or just be generally more supportive than your family.

I don't know how your girlfriend wanted you to handle this, and neither do you. Because you didn't ask her.

I'm sure I'm going to get downvoted again, so have at it relationships.

You seem well upvoted this time so I guess I will take the downvotes.