I have a real ESA, not a fake one who used an internet "doctor" to get the letter.
In any event, I only stay at pet friendly Hyatts OR I contact the hotel and tell them he's all of 9 pounds and trained. I pay the pet fee whenever there is one. Sometimes the hotel doesn't actually charge it.
I stayed at the GH Tampa and paid their $100 pet fee. I also stayed in Disney and paid the obscene pet fee of $50 a DAY.
If you're only staying one night, I was contact the hotel in advance and ask if the fee could be reduced or waived. I wouldn't talk to the front desk about it upon arrival.