No way of knowing exactly what can or can't be done to your non electrified fixture without a picture of what you have.
In general, if you have a tube to your fixture (hanging pendant) you could pull some wire through the tubing, (if chain, thread the wire through the chain)then find some sort of 12V LED that will fit in the fixture.
Keeping in mind that generally with 12V lights many use some sort of socket to plug a bulb in, you won't have a socket to work with. There are many different aftermarket 12V LEDs some may have a socket adapter with wire going to the LED.. In that case you could cut off the socket and wire directly.
Be aware, many LEDs are polarity sensitive, so if it doesn't light up. try reversing the wires.
As far as painting the bath wallboard goes, don't bother with paint. Paint is the wrong product to use for this application and will not hold up.
Instead, buy and install a plastic home bathtub wall surround, these kits are made to be universal in fit so you will need to adjust the panels to fit correctly.
You simply use adhesive to stick the plastic tub surround to the existing wallboard. Once installed, you can use Kitchen/Bathroom silicone caulk (Kitchen/Bath silicone caulk is formulated to prevent mold) to seal up the joints in the tub surround.