Hospital at Fault, Bill Now in Collections - HELP!

I have a hospital bill for a doctors visit which is relatively low ($413), which I disputed with the institution every month since I began receiving bills (over a year now). They screwed up my insurance information and got the billing all wrong. I was a college student at the time and covered under my dads plan (primary) and my moms plan (secondary). The bills have to go through the primary insurance first (get rejected because it covers nothing) and then submitted to my moms insurance which will cover the bill. The hospital reversed the process and I ended up with the bill. I immediately called and realized that they had the incorrect information on their end (including plan numbers and all---a complete mess). They then sent the bills to the insurance companies yet again. In the meantime, I continued to receive bills from the hospital, and I have called every month to dispute. They keep a log of my calls (and I of who I talk to) and keep assuring me every month that the bills will not go into collection because they have it noted in their computer database that the bills are still at the insurance companies.

However, today I received a collections notice for these bills, even though the insurance company (it's at the secondary insurance now) is still dealing with the bill. I have the $413 and can pay off the bill if need be. I only care about my credit, as I am looking to invest in a condo in the next 2-5 years. I currently have good credit with no other delinquencies.

Should I pay off the hospital and once the check clears dispute the validity of the balance with the collection agency? There is no notice of it on my credit report as of yet (all three bureaus) ---how long would one take to appear?

Or should I deal with the agency and request from them in writing that they not report it on my credit if I pay in full?

The insurance company still may pay, which is why I have been holding off. The hospital said they were too---Could the bill have been wrongly sent to the collections agency if they were operating on a computer-automated system?

I live in Massachusetts if that helps at all.

Any advice is appreciated.