hi. i'm fairly new here but had to reply to your question. i was a store manager for a fabric store for 24 years. yes, washing denim i have seen will soften with coke. i remember it from about 12-15 years ago.
an older outdoor fabric called textilene is what you sound like you are looking for in making the mesh bags. call you local home decor fabric stores and besides the outdoor decorator fabrics like sunbrella the may still carry textilene. it will work well if you can't find this pet mesh or don't want to use the window screening.