I had a similar problem with the rear reading lights. My daughter had pressed the button too hard and the light popped inside detaching from the outer structure that holds it in place.
I ended up using a suction cup with 20mm in diameter with the top portion attached to a thin wire. I attached the suction cup at the light, wire through the outer clip and while pressing the clip up, I pulled the wire down until it snapped back in. It took me several attempts with the suction cup coming off at times, but eventually I succeeded.
Best of luck with fixing your dome light!