Hello Traveller,
Question. I am not sure how to get the trim off that has the rubber insert in it.
You are on the correct track.
The lower trim is made of alumin(i)um and you can snap the strips off the white and blue clips as you already tried to. Use a flat wooden spatula or a metal one covered with some tape in order to protect the paint behind. Use the spatula from below and push it very gently between door or panel and then aluminum strip. They come easily off then.
The upper trim is made of solid brass and is chromed.
To remove them from the car you need first to pull out the rubber insert from the strips. Then you will see what the preowners did to attach the trim. From factory there were self-tapping screws/sheet metal screws. Iron screws in aluminum doors - not a good choice - a Galvanic cell. The aluminum will suffer first for sure.
In your case - as you described - other screws, most likely threaded countersunk screws (DIN 7987 or so), were used with nuts on the inside. With the rubber off you should be able to see what kind of screw they used. Or perhaps rivets...
For the short brass trim pieces on the rear fenders - if they are attached with bolts and nuts as your door trim seems to be - you need to remove the Leater/MB-Tex covers behind the seats to get access to them.
For re-assembly I used plastic screws (nuts, bolts) in order to be safe against an Galvanic cells/ or rust and to be gentle to soft components like the aluminum doors.
Good luck,
Achim