I don't know for sure if this will work so I will just throw this out there. If for some reason it doesn't meet your needs I would
really like to know about it.
https://springfieldleather.com/products/bontex-bag-stiffener?_pos=1&_psq=stif&_ss=e&_v=1.0#:~:text=Type: Bag StiffenerMaterial: BontexSheet Size: 40" x 60"Thickness Options: .015"%2C .02"%2C .03"%2C .04"%2C .05"Note: Sheet size may vary slightly. Current stock of .04" is BLACK.
I have a carry belt with something like this sandwiched between the leather. It does run continuously through the whole length, right up to the holes.
The belt has held up exceptionally well for years now.