I would think that a good automotive body shop repair tech should be able to fix this.

It depends upon the failure. If the substrate is MDF or similar and is broken/swollen underneath, it's not coming down without pulling the copper, flattening the substrate, and reinstalling the copper.

If it's an adhesive failure it may be possible to clamp it down.

Thank you! They sent me an updated image of the bottom- to me it looks like someone tried to lift w/ something and it slipped under the copper, popping it in this spot. Slight horizontal break on the MDF but what would be your first instinct to make this look better (not fix to perfection)?

FIx the edge with a pair or plers and a piece of plywood on top of the top with some weight on the "crease" would be my best thought without removing the copper. Copper cladding like that is usually quite thin and should be pliable enough.

Maybe ice or dry ice or air conditioner refrigerant on the bubble?

Arctic Freeze Refrigerant from Harbor Freight

Cold should cause the metal to contract. If it works, weight it down and leave the weights in place for a day or two. Or a week or three.