Answer: First off, every "reasonable accommodation" case is different, so I strongly recommend that you consult an attorney on your particular case.
As a general matter, HUD has said that landlords must allow emotional support animals if (1) the tenant has a disability, and (2) the tenant documents that an assistance animal is necessary to help alleviate the disability.
In your particular case, it sounds like the tenant managed to convince her doctor's office to provide her with the required documentation of both her disability and the need for an assistance animal. As such, you are probably obligated under federal law to let her to keep the pit bull.
However, one reason for challenging a reasonable accommodation is on the disability diagnosis. In other words, did the tenant's doctor actually make a verifiable diagnosis, or did the tenant just get an online "prescription" without even meeting the tenant? If that is the case, you might consider consulting your attorney on challenging the request since HUD has stated that the tenant must submit reliable documentation of a disability and the need for an assistance animal.
I have also advised clients that it is sometimes possible to remove an assistance animal if it poses a threat to other residents. However, this can' be based on the particular breed of dog, but must instead be based on the attributes of the specific animal in question. For example, if the dog lunged at or attacked another resident, you could likely have it removed.
Along these lines, I recommend that you arrange a meeting with the tenant to "meet" the dog. The purpose (which you can explain to the tenant) is to ensure that her particular dog complies with federal law by not posing a threat to other residents. Assuming the dog does not display any vicious tendencies, it complies and can stay (although any future incidents could lead to a re-evaluation). You can also require her to keep the dog on a leash, clean up after it, and otherwise adhere to any reasonable pet rules you might have. You cannot, however, require any sort of pet deposit or other fee since the dog is considered an assistance animal, not a pet.
A final issue to consider is whether a pit bull in your RV park might cause your insurance rates to rise. If that is the case, you should consult your attorney on whether that rate increase might be considered an undue financial burden on the park. If so, you could possibly deny the pit bull on that basis.
The moral of this question is that each and every reasonable accommodation case is different. Talk to your attorney before denying a tenant's request so that you can avoid a HUD investigation.