I would encourage you to respect the wishes of your friend and either attend by yourself or not attend.
I think it’s a bit of a stretch on the part of your friend to say that you and your partner attending the event would “influence others that attended to sin”. I find that really absurd, actually.
The baptism itself is a liturgical rite of the Church, open to all and not “private”. However, the social dimension of inviting people to a baptism, wedding, etc., and any party afterward may-- in your friend’s mind-- imply his/her approval of your same sex relationship if he/she were to invite you and the partner or if you had your same sex partner attend as a “plus one”.
The liturgies of the Church are open to all.
The Church teaches same sex relations are a disordered use of our sexual faculties. It teaches same sex attraction is not a sin. Acting on it sexually is.
The Church doesn’t teach that we shun those with same sex attraction or that they aren’t welcome at Church events. I think the part that has your friend troubled is you bringing a partner with whom you might display same sex affection of a romantic nature-- hand holding or whatnot-- at the event.
I think this level of concern by your friend is not likely to be shared by most Catholics. Inviting people to social engagements can be tricky for those who want to observe Church teaching on same sex relationships. How do you invite people socially who are involved with a same sex partner while expressing your desire for them to be at your occasion, but not wanting to approve of or imply they condone same sex relationships?
I think your friend is trying, but handled it poorly.