I would try again to escalate this with the bank because a code 5 implies that the HSA account has been terminated. Of course code 5 makes no sense since reimbursing yourself for a qualified medical expense paid out of pocket (or simply taking money out of the HSA for any reason) is not a prohibited transaction as defined in section 4975(c) of the tax code. Code 5 is only to be used when an actual prohibited transaction as defined in section 4975 occurs, with the gross amount of the distribution being the entire balance in the HSA and the HSA ceasing to be an HSA.
The bank indicating that you are not permitted to roll the distribution over is because you are past the 60-day rollover deadline seems to imply that the bank does not think that that distribution was a prohibited transaction because a distribution as a result of a prohibited transaction is not eligible for rollover under any circumstances.
Absent the bank issuing a correction, I would probably try contacting the IRS. There is no equivalent to the Form 4852 that you would use in place of an erroneous Form 1099-R for a distribution from a retirement plan. With Form 4852 you are normally required that you contact the IRS before filing that form, so it might be prudent to do that in this similar case as well. The IRS might intervene by contacting the bank. Otherwise, I see no alternative but to enter the Form 1099-SA using code 1, then print an mail your tax return with an explanation statement describing the bank's error and why you are not including the amount in taxable income but are instead treating it as reimbursement for a qualified medical expense paid out of pocket. Information on a 2023 Form 5498-SA showing the same account number might be useful in substantiating that the bank has not actually terminated the HSA.
