From the time your landlord accepts your ESA letter, they should stop charging any pet fees. You would also be entitled to a refund of any pet deposit. You would not however get a refund of any monthly pet fees paid until that point.

If the pet fee was paid while the animal was still a normal pet, it is likely not refundable. However, any pet deposits that were paid should be refundable.

If you provided your landlord with an ESA letter that they approved, they should have returned your deposit.

Landlords are not allowed to request tenant’s complete specific forms, and they are also expected to process a tenant’s ESA request promptly, within 10 days. This is a very odd situation where you complied with your landlord’s request but are still being denied. Nevertheless, you still need a valid ESA letter: https://esadoctors.com/how-to-get-emotional-support-animal-letters/

If the fee was paid while your animal was not an ESA then it is likely not refundable. However you would not be subject to a fee once you qualify for an ESA. You should also be aware that registering your dog does not qualify it as an ESA. Please see this article for more details: https://esadoctors.com/register-dog-emotional-support-animal/