I think the first question needs to be directed at your daughter: If you can only pick one, which one do you prefer?
She's 11. If she changes her mind in six months, no big deal.
Does the hunter barn allow her to show with lesson horses? If she's wanting to do hunt seat and this is an option, I would consider seriously pursuing this. Even if they're just schooling shows, to an 11-year-old shows are often just shows, whether schooling or rated. They will also give her good experience. And the fact of the matter is that sometimes little girls outgrow horses. I generally find that 13-14 is that "golden age" so to speak--if at 14 she still loves it, then she'll probably love it for life (and you'll be able to breathe easy that she'll most-likely always prefer horses to boys! Yay!).
So let her do what she wants, though personally I think the hunter barn sounds like a better choice since you've determined she likes it already. If she decides she wants to go back to saddleseat, then no problem. Easy enough. If in a few years she wants to get even more serious about showing, then I would consider leasing a horse or buying if it's in your budget. But wait on that until she finds where she wants to be. A saddleseat horse is a far cry from a hunter/jumper.