Suppose I use Stripe in a similar way to HIBP and a customer comes along just as my server has gone down, would Stripe try again later? I think it should wait a little while but I'd hope it'd automatically try again later as long as there's data that needs to be sync'd but isn't yet.

I can maybe imagine 403 being a "Don't try again" signal in a way that 500 or timed-out isn't.