Given that small fridges are listed as standard equipment in DVC studios (see below copied and pasted from WDW website), you should be able to get one without the hassle you had at Pop, which specifies “beverage cooler” only.
Poly DVC Deluxe Studio:
Comfortable accommodations: includes 1 full bathroom (2 showers), kitchenette with small fridge and microwave, and a porch or balcony.