Most cheap-ass hollow doors have a lot less structural timber than that door - 1½" max top and bottom, about the same on each side, and a block where the handle goes. ... Try picking one up - they're shockingly light.

A window would be relatively difficult, time-consuming, and expensive (it will need a custom size), not to mention that it will almost certainly look weird due to size and/ or location with respect to the floor and/or other windows.

Take measurements before you buy it, then cut it to size in the parking lot outside the store where you bought it. ... I have been known to take a T-square and a handsaw with me to Lowe's to cut timber or boards even though I own a pickup. Sometimes it is just that much easier to carrying material already cut into smaller pieces.

Lots of potentially good answers already, so I will try to keep my (initial) post short: before you can do anything to address the problem you must determine whether you have a leak from outside or have a condensation problem inside. ... I am assuming you have considered and discounted a plumbing leak.

If it's a leak from rainwater the problem will likely be year-round. If it's condensation it will likely be worse in winter .... unless you have air-conditioning which can, under some circumstances, cause condensation issues in the summer.

So solutions- if it's a leak, any plan must start with stopping the water entering the structure from the outside. All the interior wall treatments and dehumidification will be for nothing if the moisture is still entering the wall cavity, as it will just cause mold and rot.

If it's condensation, then running your AC will help, or a dehumidifier if the problem is only in one room. Also move furniture away from exterior walls to allow air to circulate. A fan (rather than a dehumidifier) might solve the problem if it's minor. In the winter an auxiliary heat source, such as an oil-filled radiator placed near the damp spot should also solve condensation dampness. .... When I was a child, I used to have a book case at the foot end of my bed, up against an exterior wall. We found mold had formed on the wall behind the bookcase, due to coldness and poor air circulation. Swapping the bookcase with a small radiator that already heated my room, entirely solved the problem. ... It also helped keep my feet warm! 😀

Considering that it's a bathroom, so presumably not a massive space, you might at least try using a reciprocating saw with a scraper-blade tool to remove the glue.

I put strip LED lights under all the wall cabinets, so they put more light directly down on to the work surfaces. I wired them all together in a junction box, connected a rocker switch under the cabinet, and ran a cable up to the top and along to the double receptacle for the microwave.

Then I did the same with upward-facing LED strip lights along the top of the wall cabinets..... and connected them to the same double rocker switch under the cabinets.

Whole set-up uses standard electrical connectors, boxes and switches, but is still a "plug in" set-up, consistent with the strip lights design as plug in fittings.