"spans into 2023 and is reported in 2023"

The tax year of the contributions isn't important.

"I have a customer who contributed $6K to an IRA in 2022 and did a backdoor Roth IRA in 2023."

By definition, then, that is not Back Door.

A "Backdoor" is just a conversion. What makes it "backdoor" is that the contributions are supposed to have been post-tax, so that makes them basis (and nondeductible) and the conversion is nearly simultaneous to that contribution date, so that it avoids any gain (which is untaxed income). And the other trick is that this person is supposed to have no other Trad IRA, SIMPLE IRA or SEP-IRA pre-taxed or "never taxed" funds, because these get aggregated into applying Basis to a conversion to determine if there is pro rata taxable conversion.

In other words:

Everything goes in as basis and is immediately converted, and is the only amount that existed in this type of account. That's the magic of "back door" because it is how a person not eligible for Roth, still manages to end up with funds in Roth without triggering a taxable conversion event. Ta Da!

So, set aside "backdoor" and simply determine if all of this (all in one tax year) conversion is a taxable event, a pro rata taxable event, or a nontaxable event.