I may be waay out of line, but the fact that the catheter broke inside your cat sounds like much of this problem is the vet's problem.
I worked in dentistry for over 25 years. A couple of times a dental instrument broke inside a patient's mouth or tooth. We did not charge the patient for replacement instrument and did not charge the patient for the extra work to remove said objects. Once a patient swallowed a brand new crown before it was seated. We paid for x-rays and Lower GI Exam as well as made a new crown for free. Similar situation with a patient who swallowed an implant component. We paid for medical care AND replacement components.
I don't know if the same holds true for veterinary care or for catheters. But I'd sure be suspicious of the bill and would want to talk to the vet - not just the business manager. I know next to nothing about kidney stones but don't understand how (1) the vet did not know that one was still present, and (2) the stone 'broke' the catheter.
Sounds odd to me.
I hope your cat survives. I miss mine.