It may well be that they have the legal right to do that, but I'm surprized that they'd actually do it. With all the folks who don't pay a penny, I'd think they'd be pleased as punch to have someone actually paying it off. DW and I had medical bills a few years ago, before we had insurance, that we were able to pay off over time. Of course, that was an ER situation, and we told everyone the situation as we were entering the doors, so they were very helpful.
I'd try and negotiate something with them, and if that doesn't get anywhere, go over the heads of the folks you're talking with and straight to whoever runs the hospital. Since the vast majority of your bill has already been satisfied, you're not exactly a deadbeat.