You can only take the claim of right credit on your tax return for the year that you repaid the income. You can't file the credit separately. So on your 2022 tax return, that you file in 2023, you can get the claim of right credit for the amount that you repaid in 2022. The credit will be included in the refund from your 2022 tax return, so you won't get the money until sometime in 2023. If you are waiting for that before you repay the remainder, you won't be able to complete the repayment until 2023. Then you would have to claim another credit on your 2023 tax return, that you file in 2024, for the amount that you repay in 2023. (Calculating that credit on your 2023 tax return would be extra complicated because you would have to take into account the credit that you already claimed in 2022.) If the remaining amount that you repay in 2023 is $3,000 or less, you would not be able to claim any credit or deduction for it.
Also keep in mind that the credit is just an approximation of the income tax that you paid on the amount that you repay. It is not the full amount that you repay. Furthermore, you do not get a separate payment for the claim of right credit. It's just applied as a credit towards your total tax for the year that you claim it (2022). Depending on your overall tax situation, the refund on your 2022 tax return could be less than the amount of the claim of right credit.
My gut feeling is that your proposed plan is not a good idea. It's too complicated, and it delays the final resolution of the situation for a year. I would suggest that you do whatever you have to do to complete the repayment this year.
