I decided to get to the bottom of this. I have saved about $1000 from tips on this forum so I gave it a go:
I removed a chrome door handle insert for paint!
** They are plastic and may break. I don't know if you can even buy them without an entire handle a$$embly so proceed at your own risk!! **
All you need is a small flat screwdriver and a piece of dental floss or fishing line. The front has a smaller hook than the rear. There is adhesive. A small amount of Goo Gone or WD40 will remove the adhesive but you will probably need new adhesive to reinstall. I removed it without chemicals.
Gently pry the FRONT of the insert up. Slide the fishing line or floss under the insert and use it to break the adhesive from front to back. Use the screwdriver to lift as needed but BE CAREFUL!