Hi,
A good contact glue intended for automotive use and thus able to resist high temperatures should do the job. Either in spray or brush on version.
Würth has a spray version that is intended for this type of use, the extra strong version is what I've used with good results so far. A bit expensive but probably worth it.
My Pro upholstry guy never uses anything but brush on glue, but I find it easier to use spray for gluing carpets. (use according to instructions: apply to both sides, let dry so it doesnt stick when touching, then join the surfaces.)
I am just surprised to hear that the carpets came off the insert boards.
I have never been able to remove the old original carpet from the seat back inserts
without destroying the boards, they are just that impossible to get the carpet pieces off the boards.
I always ended up having to cut new boards for this.
Hans