I guess I'm going to go against the popular opinion here and suggest trying both the tape the glue route first.
I had something similar once --but not identical-- on a trim molding on a vehicle and on the advice of a friend who worked at a body shop for a time, used the 3M double-sided tape and had no problem with it coming off again.
3M™ Super Strength Molding Tape
Through Amazon:
3M™ Super Strength Molding Tape
I have also in the past used J-B Weld on an RV where a portion of laminate in one section at a corner was loose and used just a small amount of J-B Weld SuperWeld near the corner and then to hold it down, used several pieces of no-residue duct tape or gaffers tape to secure it for at least a day. When I removed the tape, it seemed like it adhered perfectly with no sign of it being loose. Again, never had a problem with it coming undone again.
At Amazon:
J-B Weld 33120H SuperWeld Glue - Clear Super Glue - 20g
At Amazon:
Industrial Max Strength Duct Tape, No residue (Silver)
I might try either of those before a pop-rivet but that's just me. A rivet seems "too permanent" to use where I might desire a cleaner look. I've used rivets in places where aesthetics don't come into play but that trim piece looks to be right on the front cap (?) where I'd prefer not to see a rivet if possible. Of course if tape or glue does not work, then I suppose you'd have to revert to a rivet.
(good to hear you were able to reach Summit and found them helpful. Were they able to actually solve your problem?)