I would put a patch of plastic needlepoint canvas on the backside to not only stabilize, but also give you a structure/pattern for your mend.
Shoot, I was going to recommend this Darice that I got as pretty hardy but maybe not strong enough for what you're looking for? But I'm pretty sure I couldn't rip this without scissors. I made Christmas ornaments: Darice Perforated Plastic #14... https://www.amazon.com/dp/B001145310?ref=ppx_pop_mob_ap_share
https://www.amazon.com/Darice-Perforated-Plastic-Canvas-White/dp/B001145310/ref=sr_1_1?ie=UTF8&qid=1547516663&sr=8-1&keywords=plastic+cross+stitch+canvas+14+count - try this link. i found it off amazon when i searched for "plastic cross stitch canvas 14 counts." i wish you luck on your hunt
I used this! It’s comparable to 14ct Aida.
Darice Perforated Plastic #14... https://www.amazon.com/dp/B001145310?ref=yo_pop_ma_swf