I misread the 2022 Form 8606 that you posted. I think I saw the $6,000 on line 5 and thought it was on line 4. The form does indicate that the $6,000 contribution was made in 2022, so ignore what I said about it appearing that the contribution was made in 2023.
In your original post you said, "... the amount recharacterized was $5,231.48. The amount converted back to the Roth was $5,272.30." However, your Forms 1099-R show that the amount recharacterized was $5,272.30 and the amount converted was $5,231.48. Given the correct information, your 2022 Form 8606 is correct, so no need to amend that. $769 is the correct basis that carries forward.
Check line 3 of your 2022 IRA Information Worksheet. If it shows zero, this is likely where the problem lies. Before the Tax Cuts and Jobs Act of 2017 suspended miscellaneous deductions subject to the 2%-of-AGI floor through 2025, one could take a deduction for unrecoverable basis. TurboTax would automatically include this deduction on Schedule A and zero the basis when the year-end value in traditional IRAs was zero (although in my opinion this deduction should be optional). With the suspension of this deduction, TurboTax no longer puts the deduction on Schedule A but I think it still zeros the basis that carries forward.
I would enter $769 into 2023 TurboTax as your basis from 2022, enter as your explanation that there is no actual adjustment to your basis and that 2022 TurboTax mistakenly zeroed your basis carryforward despite $769 being present on 2022 Form 8606 line 14.
(If you enter at least $1 as your year-end value in traditional IRAs, I think that TurboTax won't zero your basis.)
