"Code J is entirely custodian's fault."
Presumably you had the opportunity to review the details of the transaction prior to finalizing the distribution request, so it's doubtful that the custodian made the distribution as an ordinary distribution without your approval.
If you do want to challenge the accuracy of the 2023 Form 1099-R, using 2022 TurboTax you would have to enter a substitute Form 1099-R (Form 4852) describing the error and your attempts to get the erroneous Form 1099-R corrected. With that done, you would report the substitute Form 1099-R on your 2022 tax return as you described. You would need to file the same substitute Form 1099-R (Form 4852) with your 2023 tax return or 2023 amendment, otherwise the IRS would determine that you failed to report the $154 of income shown on the code-J 2023 Form 1099-R.
Given that the penalties are only $13, I would just treat the distribution as an ordinary Roth IRA distribution as reported by the custodian and as I described in my post above. It would be less effort (no Forms 4852) and essentially no chance of the IRS disagreeing with the way that you report the distribution. The $103 of gains would end up on the 2023 tax return instead of on the 2022 tax return. No matter what you do, both the 2022 tax return and the 2023 tax return are affected, so if both of these have already been filed, both need to be amended.
