"Then, I was told that I could transfer the $177 back to them and then submit a “distribution of excess contribution form,” like I hoped. "
I think you should declare victory here, and do it this way. Yes, it's not the same account (but at least it's the same HSA custodian), but what is important here is that your tried to correct the situation as best as you could despite HSA custodians being notoriously reluctant to make any change out of the ordinary.
"do I just make sure that I under-fund my 2023 HSA by more than $177 under the limit," - Yes, this is the second thing to do, showing that your excess was properly discharged in the subsequent year.
Just document everything you did and why and stick it in your tax files in case anyone ever asks.
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