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Call or go to the hospital and have them recall the debt in exchange for payment. That's it.

If it's already been reported to the bureaus, there's nothing you can do about that for the moment. Once the hospital recalls the debt, the collection should be removed. I would just go in to the hospital to get a receipt, just in case.

While not enacted into statute, the CRAs entered into a settlement agreement with the offices of the AG of several states back in May 2015, titled:
“ In the Matter of: Equifax Information Services LLC, Experian Information Solutions, Inc., and Trans Union LLC
Assurance of Voluntary Compliance / Assurance of Discontinuance”
That settlement agreement includes the following specific provisions related to medical collections, which are now legally bindng upon the CRAs:
“3. Medical Collections
consumer's individual payment obligation is for a medical account, the CRAs
shall prevent the reporting and display of medical debt identified and
furnished by Collection Furnishers when the date of the first delinquency is
less than one hundred and eighty (180) days prior to the date that the account
is reported to the CRAs.
special comment codes of "BP" for debt identified as "paid by insurance" and
"AB" for debt identified as "being paid through insurance" and instruct
Collection Furnishers to remove or suppress medical accounts reported as
"paid by insurance" or "being paid through insurance" if such accounts were
in fact paid in full by the consumer's insurance carrier and were not the
obligation of the consumer.
suppress known medical collections furnished by Collection Furnishers from
files within the CRAs' respective credit reporting databases when such debt is
reported either as having been paid in full by insurance or as being paid
through insurance. “
Thus, any reported medical collection will not be posted to your credit report until after 180 days from DOFD, and even if then posted, will be removed by the CRA if shown to have been paid via insurance.
Are those provisions satisfactory in meeting your needs?
If so, you can rely upon them to have any posted medical collection removed if added more than 180 days from DOFD, or if it is thereafter finally paid via insurance
If this happened a year ago, they didn't immediately send it to collections. Your title is misleading.