@meupatdoes has a good point, that it almost doesn’t matter what the contract says right now, because either party has the right to propose new terms and renegotiate. All of the below assumes that the OP doesn’t have actual ownership of the horse, and it’s more of a lease.
The OP (well, whichever adult signed the agreement) holds the far better hand here, because she can walk away ( following any/all termination requirements that are in the current agreement, of course, such as 30day notice). There are lots and lots of horses out there. I’m not saying this would be easy in an emotional sense, but from a negotiating standpoint, it’s a good position to be in.
In contrast, the barn/former owner is faced with two options, both bad for her. He options are to 1) let the horse be moved and lose the board income, or 2) don’t let the horse move and the OP walks away, BO still loses the board income, AND now has a new horse to feed, shoe, and care for. So it’s in the BO’s interest to agree to let the horse be moved, and through gentle, polite negotiating, you can help her realize that (in a way that lets her save face).
So it may be worth the attempt to negotiate-- but ONLY if staying at this barn is bad enough that you’re willing to give up that horse.
ETA: the above is more theoretical about the right to renegotiate, and it’s not direct advice to the OP. We don’t know the Rescuer/BO’s side of the story.