I always have to look this up.  First I'm going to pick on you about terminology.  This doesn't sound like a recharacterization.  That's when you get to "rewrite history" and pretend this went into a Traditional IRA instead of a Roth (or vice versa).  Sounds like what you have is a corrective distribution.

If you're lucky you can get the broker to tell you what 1099(s) they're going to issue and what amounts and codes they're going to put on them.  If this is a 2023 1099-R with a code P for $6,166 / $166 then go ahead and enter it in 2022 so you don't have to amend later.  As Michelle said, there might be a penalty on the earnings.  I *think* that's what I would expect (because the excess contribution was made in 2022) but peruse the 1099-R instructions to see if you can figure out if this should be taxable in 2022 or 2023.

The other alternative is a 2023 1099-R with a code 8 which is taxable in 2023.  I *think* that's what happens when you make a Jan 2023 excess Roth IRA contribution but designate it for 2022, then fix it in March 2023.  But it's April 6th and I reserve the right to be remembering this all wrong. 🙂

Rick