The current version of the instructions for entering a back door Roth IRA conversion makes note in step 11 that if none of your Traditional IRA contribution is eligible to be deducted, then you will not see the question about making part of it nondeductible. The default is that all of it will be considered nondeductible.
Once you have entered the non-deductible Traditional IRA contribution as part of your 2022 return, you are done. You will enter the Form 1099-R reporting the distribution from the Traditional IRA and subsequent conversion to Roth IRA as part of your 2023 tax return. The distribution and conversion occurred in 2023, not 2022.
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