Here is a picture. I am still trying to find the one we bought. This one is on ama zon. I searched refrigerator storage.

This one is from office depot

Not exactly what you have in mind, but but, we run an inn and have a mini fridge in every room. I didn’t like the look of them either, so we bought microwave carts that house a fridge. Ours are made of wood so they blend in with the New England vibe of the inn.

These two are from ama zon. search mini fridge cabinet for bedroom. This one doubles as a bedside table.

Whatever you end up finding (or having built), I'd suggest using a medication themometer to monitor the fridge temp and to detect hi/low temp excursions for your particular medications.

I'd look into furniture manufacturers that cater to the hospitality business. Since you own an inn, you are likely eligible to purchase from them. Do a google search and call up a few.

https://www.wayfair.com/furniture/pdp/sand-stable-liliana-65-wide-2-drawer-sideboard-with-refrigerated-beverage-cooler-w003333550.html?piid=1792600457

https://www.hotelsurplus.com/74-credenza-with-built-in-safe-and-mini-fridge.html

https://www.pinterest.com/pin/157555686937265340/

https://www.newformsdesign.com/en/products/hotellerie/minibars-hotel/minibar-40-liters-drawer_20461.php