There is this really cool stuff, fusible vinyl for just such projects. It doesn't make it waterproof exactly, more like wipeable. I actually have some. When I first bought it I wondered how heavy it would make my project feel, well, it didn't. The fabric remains flexible and not overly heavy.
Also, there are many weights of clear vinyl at Joann. Could the bag simply have a loose vinyl liner? You can use tissue paper over the top of it to sew, if it's a problem moving over the pressure feet.