You can ask that the mini bar be emptied, and you pay the $15 a day to use it.
If you buy a mini fridge and bring it, that would also work, but a mini fridge is about $100, so you'd probably break even on the price, unless you're staying for a week or more.
We bring 2 coolers with us, and I keep filling them with ice at the hotel. A stocked cooler will actually last a couple days once everything inside is cold. I dump the water out once a day, and add 1 bucket of ice.
A styrophome cooler costs about $7 in the beer isle of Publix. I'd buy two of them, then when you check out leave them near the ice machines.