Should my SO baptize his cousins baby alone?

I need some opinions. My SO’s cousin asked him to baptize his son. And no not me, We’ve been together 18 years but I’ve never really talked to his cousins girlfriend because I’m shy plus I’m usually inside watching our kids when we go to parties. Anywayyy I’m fine with not baptizing the baby plus we are Christian (they are non practicing Catholic). He will be baptizing the baby with the girlfriends sister. Lately my SO has been thinking about declining the offer because 1. We aren’t catholic 2. he feels awkward, (especially all the picture taking with another woman) and 3. Him and his cousin aren’t really close anymore.

I think it’s a bit too late to say no. Although they don’t even have a date set or anything yet. But what would you do? Help us please!