I mean, it comes down to whether you decide to side with eBay policy compliance or honor your arrangement with the other buyer.
Technically, and I think you know this, the eBay buyer takes precedence, even though the original person will be disappointed. That's a risk you both took when you decided to re-list for an eBay purchase rather than taking the transaction private at that point, which you could have done, as the original listing ended.
The other buyer did buy it fair and square. I feel ya.
If it were me, I'd probably message the eBay buyer and explain that this was a custom link for a person who had already arranged the purchase price and they made the BIN in the two minutes while you were messaging the other buyer. I'd give the eBay buyer an option as to whether they would be willing to cancel the transaction or whether they're set on getting it. The first buyer could have purchased during the original auction...
It is a sticky situation, for sure. I think if you handle it with grace and are honest about the situation, you can manage it with a satisfactory outcome. I would just put the first buyer on hold for the time being and see what you can work out with the second buyer. No need to alarm the other person if the BIN person is willing to cancel the transaction.
My guess is that I think the second buyer will be bummed but, (especially if they're active in the collector community) will also understand that these things happened and they will appreciate your honesty and agree to cancel the sale - IF you approach it like this and don't panic everyone