Well, I managed to get a chance to try felting the bag in a top loader washing machine. Put a few pairs of jeans in and a towel (prob not a good idea), and the bag and straps were in a fine mesh bag. I set the machine to hot water and put some gentle soap in. I did not manage to catch it before the spin cycle but it seems to have been fine.
The bag shrank more and the fabric is thicker and stronger. Still a bit floppy but much better. I attached the straps using nylon thread and backed the attachment point with a button on the inside. I don't have a closeup of the felted fabric but here is the bag:
Cool and much closer to the picture of the finished product in the pattern book.