Worth a try, but before I did this I would verify continuity of the switch when pressed. I say this because I'm assuming the trunk solenoid is bad.There is an easier way... Pull the trunk release button out where it plugs in run a wire from the positive on the battery to where the plug goes into the release button. Just run the wire around the outside of the car this is just to feed temp voltage to the button. While holding the button down touch the wire from the batt to the metal post and it'll feed power to the trunk release solinoid and as long as the selniod is good the trunk will pop as if you had the key in and on. Most of those buttons just pry gently and it'll come out.
No visible rivets. On the underside where the license plate lights are, there is a plastic cover between the lights held in place with one rivet. Might this be what you are speaking of? If so, I could probably get to the rivet with a right angle drill. Or better yet a dremel would do the job of rivet removal...IIRC on Tbirds there are rivets you can drill/punch out and take out the lock cylinder clip and lock cylinder.(cougars too but the reflector panel is in the way) Otherwise my method is exactly what @1997ThunderbirdLXV6 described; pull the back seat crawl through the trunk and unbolt. Thrashing the lock cylinder doesn’t work.