Thank you everyone for all of these suggestions! There seems to be little consensus here and from my other research on this question. From the residue on the mat and from my research, I concluded that the manufacturer originally (20 years ago!) used a double-sided tape, and 3M makes such a tape designed for attaching trim to cars, so that is what I decided to go with:
3M Automotive Acrylic Plus Attachment Tape 06384
I will re-post to update on how it worked out!