I would love for the Fat Cat Bank to be a usable piggy bank.
The functionality might be as follows:

(1) If the gil on you is greater than the "Spending Money" set amount, deposit the excess into the Fat Cat Bank

(2) If the gil on you is less than or equal to the "Spending Money" set amount, open a menu with these choices
(a) set "Spending Money" amount to a value greater than or equal to 0
(b) deposit/withdraw gil from the Fat Cat Bank