First, I'm so sorry about the circumstances of this situation.
I think you are between a rock and a hard place. You need the expertise of the lawyer(s) you hire in order to preserve the monies of the estate the best way you can.
I don't know the specific term (?trustee), but it would seem to me that appointing an impartial person to oversee the money in the trust might save a lot of hassle, disagreement and heartbreak between you and baby's mother in the long run if both of you are appointed to oversee the monies together. You obviously will see things very differently - as you already are. Baby's mom will probably be more interested in the daily expenses NOW of raising the child, whereas you will undoubtedly will be seeing the "bigger picture". Unfortunately, this is what happens when teenagers have babies. Baby's mom is foolish if she's not applying for assistance for which she qualifies, but it sounds like she's very immature (and estranged from the support of her family to boot; she is naturally going to go to the company of 16 year olds who show her attention if she has nowhere else to go). OTOH, I'd imagine she might need more than one can of formula or one box of diapers from you from time to time.

I have to ask, are you trying to prove a point to her about this? If so, you are headed for a very long road in the life of this child with the mother who's the "gatekeeper" to her. I very much sympathize with your situation,

but you need to find a way to get along.
My advice (FWIW) would be to do the best you can, then let it go as much as you can. I highly doubt that, unless you are appointed the sole guardian of the money (which likely isn't going to happen), it is going to last through to "college" or otherwise long term. The best you may hope for is that an impartial person could help it last as long as possible without your needing to be involved in that aspect. Then maybe you could concentrate on just being Grandma.