I have used flat rate USPS envelopes alot. Since the rate is the same, can you put cardboard inbetween the fabric and squash the air out of the envelope before sealing it down.
USPS rules do state that you can't alter the package shape. I would weigh your stuffed envelope before you get it to the PO. Even at 13 ounces first class is only 3.26, but not packaged in a priority mail envelope if is to be mailed first class.
You can purchase large kraft style manilla envelopes in a pack in the dollar store and use them provided your pkg is under 13 ounces. Once its a pound or more package, parcel post can also be a cheaper way to ship something. Its slower, but cheaper.
You can get yourself a small electronic (batteries or electric) postage scale at Walmart, Office Max, Staples and Office Depot. They may help you out. I always weigh stuff before I send it. The USPS rates are free to download. Hope this helps you out. from, Debbie, Phoenix, AZ