yes, it's good that he's trying to take responsibility for his child, but when it's out of spite the actions is meaningless and rather pointless, just like it would be ifshe tried doing it to him...not every female throws a bitch fit about their kid's father not being in their life...but you don't have to have FULL custody to be in your kid's life either
but yeah, the U.S. court system favors children to be with their mothers, especially at a young aGe like your son is, just keep that job you have and keep saving money, don't do anything to where the courts can say anything bad about you and you're good...offering him stuff so he'd stop the proceedings might come back to bite you in the butt though, so try not to say anything about that if you're testifying on the stand or something...
let them know that you feel he's just doing this out of spite, that yes he might care about your son but he doesn't have to have full custody to be a father to his son...if you can't get full custody then get shared custody where he can have him for the weekends or something
good luck with this though ma