It sounds like the HSA administrator is confident that they will straighten out their internal paperwork and don't see the need to correct the 1099-SA. In my experience, it is difficult to get such administrators to change their internal processes.
What I would do is to document everything about this situation for both years and keep them in your tax files (with references from one year to the other) in case anyone ever asks. I have not heard of audit letters on HSAs, but it has to happen sometime, so just be sure that you are ready to tell the IRS (if necessary) what happened and why your tax returns are different from the HSA administrator's paperwork.
So long as you mark these distributions as for medical expenses, your tax due should not be affected. (your 8889 will be different however, and this is what you should be prepared to explain).
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