How to backlight a faux window?

Battery operated fairy lights might work for you.

A friend of mine built a simple shadow box frame to fit the window (actually, she was using a poster to highlight). Drill or cut a half circle at one end of the frame for the cord to rest in so the box frame can sit flush to the wall. She mounted LED lights all around the inside of the frame, bringing the cord down through that hole and into the outlet.

I can't remember if she put a backing board on the frame, but thinking of your project, I think I would not only do that, I'd then glue a mirror to that backing board to really bounce the lights through your window.

If the shape of your window doesn't completely fill the frame, you could top your frame with a sheet of glass or acrylic and position your window over that, if necessary.

I'd love to see a photo of your finished window project!

And if you don't have an electrical outlet near where you want to hang it, you could always go with battery operated lights, in which case I'd definitely go with LED lights. Maybe make a hinged "door" at the least visible area of the frame so you can replace batteries without having to remove it from the wall.

Good luck!

They make lights for paintings that would work.

How about white Christmas lights

Whatever lights you use, I would put a sheet of clear plastic that has been lightly sanded in front of them to diffuse the light so the lights are not just bright in spots.