Congratulations on finishing your kit!
I think you’ll have difficulty finding a paint that adheres to the TPU and is flexible enough that it won’t flake off. Even flexible polyurethane paint can’t handle the folding (I’ve tried it).
The Seam Grip plus pigment idea @legslegum mentioned will probably work.
A permanent marker could also work.
You could cut out the name from scrap fabric and iron it on like I did here: https://www.diypackraft.com/2019/10/08/a-diy-packraft-logo/ These iron-on (fabric) logos have held up very well, but cutting out all the little bits was a pain.
I also ordered some iron-on logos from an online sticker-making website and they worked okay, but it’s easy to mess them up while ironing (stretching, wrinkling) and if you don’t do it right they will peel off.
I got some “permanent outdoor” vinyl stickers made as well, but I haven’t abused them enough to know whether or not they’ll last.
Cheers!