J.M.
I was so worred like you and held off introducing my boyfriend for 4 months(- a little longer). I actually went to my ex's therapist with my ex to tlk about it because my ex was so agnst it and I didn't want there to be bad blood. The therapist agreed with M. that it didnt matter as long as I kept him as a friend and no PDA until I knew it was going to be a long term relationship. He also said some people know it's going to long term in weeks and some take months.
I introduced him as a friend in March and by the end of the summer we were begining to hug in front of her and that progressed into the affection you'd see from any couple in front of kids, and then moved in together in the fall.
If you wouldnt hesistate to let another friend join game night, invite him. Treat him as a friend. Its kind of fun sneaking hugs=)