Here is your dilemma. if you take down those fixtures there will be two holes in the mirror, that being said, they will be up high and most likely off center. So, to get new lighting that is not stuck in the corners, and covers the holes behind it, might be next to impossible unless you run a thick bar that covers the space. But some times back in the day they only drug wire over and did not put in actual junction boxes, so the bringing it to code might be another issue. 1. take one of the lights down and see what is going on 2. probably have to replace the mirror and move the electrical location.

You should be able to find choices that are 22" or 24" by 28 or 30" which would work for the first room. 24" X 36" is another common size (for the second room), but since you only have 37" I'm not sure that would work without being able to see the light fixture. Can you post pictures of each room? That's not to say you wouldn't be able to find something out of these size ranges. Use those as rough guidelines when searching and see what's out there that works stylistically. Once you find something you like, you can use painters tape to put up an outline in the size you're considering buying to help yourself visualize it. Most of these things are returnable too if it ends up not working out.

Why do you think you need MDF backer? Are you going to hang the mirror with French Cleats? Typically commercial would install that way. Residential is glued directly to the substrate.

Mirror on mirror Super Slim LED Digital Alarm Clock, Mirror Surface for Makeup, with Diming Mode, 4 Levels Brightness, Large Display, Easy Setting, Dual USB Ports, Modern Decoration for Home, Bedroom Decor, White https://a.co/d/d9TmgCC