How do I enter into TurboTax an IRA withdrawal I made in late 2022 and redeposited into the same account in early 2023 within the 60 days?

Don't worry about putting it back in the next year.  After you enter the 1099R it will ask what you did with the money.  Say you rolled it over to another account even if you put it back into the same account.   If there was withholding taken out you needed to replace that with your own money.  Or the withholding becomes a distribution and taxable.

Although I said I rolled it over to another account, TurboTax is telling me the "IRA contribution" is not permitted because the amount exceeded my earned income.

Huh?  Why are you entering it as a new contribution under deductions?  Only enter the 1099R, nowhere else.  It is not a contribution, it is a rollover.