The page on US passports for children under 16 is here. Normally, the child and both parents (if both have custody) need to appear in person. If one or both parents cannot attend, then the parent(s) who cannot attend can have a DS-3053 notarized, to be presented with the child when applying. I don't think it's possible for the child to not appear in person at the time of application.
So I think what would normally happen when parents live apart is that the parent who doesn't live with the child would do DS-3053, and the parent who lives with the child would apply for the passport in person together with the child. But in this case, you don't want the parent who the child lives with to apply for the passport (because you don't want the passport to go to him, and because he doesn't want to pay for it).
So I think some possible options are:
- You go to where the child lives, and apply in person with the child and the father, and you can pay and put your address on the application. Or
- If the child visits you sometimes, you can have the father have a DS-3053 notarized, and you will apply in person with your child when your child visits you.
