USPS flat rate envelopes are NOT documents only. Here's what the DMM rules are:
Flat Rate Envelopes and Boxes
Any amount of material (up to 70 pounds) may be mailed in a USPS-produced Priority Mail Flat Rate Envelope or Flat Rate Box. When sealing a Flat Rate Envelope or Flat Rate Box, the container flaps must be able to close within the normal folds. Tape may be applied to the flaps and seams to reinforce the container, provided the design of the container is not enlarged by opening the sides and the container is not reconstructed in any way.
As long as you aren't cutting the envelope up and retaping it to fit your item, you can use the flat rate envelopes. You (or your customer) might need to print this out and ask to speak to a supervisor if a counter employee gives you a hard time.