After talking with Kara and HonorHealth further about our particular situation, they took responsibility for what had happened (which is a lot more than I can say about United Healthcare), and HH went above and beyond to assist with the remainder of the bills.
I am really happy that management is taking notice and actually doing something about it.
I hope future changes and improvements will prevent this sort of thing from happening again, and wish HH good luck and speed on implementing the changes needed.
A big thank you to HH!