The time to introduce yourself to this family as the girlfriend is NOT at the party - friction or no friction. I wouldn't send a birthday gift to the child because he won't know who you are and why you are doing it. You're not a complete stranger but you've not been in the "gift giving circle" up to this point, so don't use the child "Liking him through the stories" is not enough right now. It's not up to you to "make yourself known" unfortunately - I totally sympathize with your desire to be a part of his life since you are now together after being apart for so long. But even you admit there are some "adjustment bumps".
I think it's great that you support his relationship with his nephew and that he is not abandoning this child just because his marriage broke up. That's very admirable. I would continue to support that. And you seem to be sympathetic to the fact that he is going to have to run into his ex at various points throughout the year.
Give yourselves at least another few months to develop a relationship with this man, and to get used to doing things together and being together rather than just being email and Skype friends. That's a beautiful thing to encourage. Then, when you're both ready to meet his "family", you will have a really secure bond and the two of you will not be awkward in any way. You will project an aura of confidence and togetherness that will be evident to others, and they will be more likely to embrace you because the two of you will make it so comfortable for them to do so.
When I started dating my husband (divorced, 2 kids), we wound up renting a house together. Still, I moved out on weekends when he had his kids for a while - it was a huge hassle BUT it gave them time to be together in this house and for them to see him as a single father with a home (he had been living with his mother before that, so the kids only went to Grandma's house to see their dad, in their eyes.) After a month or so, I joined them one night for dinner at a restaurant with his mom and the kids. Another time I went to the park with them. I never went to any big family things or any "occasions" at first. It was a problem (inconvenience, my feeling left out in some ways) but I had an underlying confidence that things would work out for us. When we did start to do things together, people saw how happy he was (rather than him feeling awkward about any family drama). But we started out doing things on OUR terms and on neutral territory rather than "loaded" events like holidays and birthdays and so on.
If you are willing to be patient for another 3-6 months and start gradually, it might be better for you and it won't drive him away. But discuss the possibility and a sort of tiered "schedule" for being involved, and say things like "I can't wait to meet this child" without saying it has to be this coming weekend, you know? I don't mean to suggest in any way that you allow him to keep you in a closet and never do anything until he's good and ready. But you guys are not really "together" yet in terms of being used to each other. Get that part down first, and the rest will come.
Good luck!