How about a needle board? Mine is old and much loved. Gotten me out of a number of wrinkled troubles and protected the napped, looped, felted, thick, etc. fabrics in the process.
i suspect instructions are still included with the boards. Failing that, there're always the kind folks at Dritz or maybe a Vogue sewing book for a bail out.
It might take forever to get rid of wrinkles in some fabrics but if nothing less onerous worked, ....
in the meantime, you have our good wishes and please share with us what finally works. I am not among the most persnickety folder up-ers and putter away-ers and your research will very likely be very important.
Alvie