J.P.
If you invited my child to go, I would neither send my kid without any money to pay for his own food nor would I send him at all if the adult-to-child ratio were too few adults to too many kids. I'd insist on going and helping you out and I wouldn't think twice about paying my own way. In fact, I'd probably offer to help out with the kids' tickets. I'd like to believe your child's friends' parents would react like I would.
I also like the idea someone else suggested of creating an "admit one" invitation. If you know the other parents, just shoot them an email saying "Hey, we'd like to invite you all to come with us to the baseball game. As a birthday treat to himself, (your son's name) wants to treat your kid to his way into the game."
Good luck, and birthday hugs to your little one!