Interesting delam issue - revised 6-24, more pics

I think anev942 has the best idear.
I think it would be hard to ensure enough adhesive coverage from a few very small strategically placed holes to endure it stays up.
I “think” it’s heavy plywood sheeting but I’d verify that and use some trim strips. Push it back in place like you’re thinking and sink a bunch of screws in it.
Something like the pic he showed or possibly nice looking washer and screws. Stainless or coated and make it look like it supposed to be ther.