This is an update to this post.
I turned the backpack inside out to see the damage on the inside. Most of the black plastic coating was gone, and you can see the patch I applied in the outside.
I decided to coat pretty much the entire bottom with Shoe Goo sealant, like I did in my last repair. This gave the material a lot of extra stiffness, as well as waterproofing it. You can see it applied in the shinier areas:
Here is the sealant on the inside of the biggest rip, holding the patch in place:
I know this update isn't particularly visually impressive, but it felt like a pretty successful fix. I'm looking forward to being able to use this bag again!
Thanks! Good luck with the rest of the repairs.