Basically, we don't know if the credit will work the same way for you as it does for a US resident taxpayer, because we don't do a lot of international tax issues. But you should definitely try and claim the credit on a form 1040-NR that you will file for tax year 2022. You would file this return between January 31 and April 15, 2023.
Note that Turbotax does not prepare form 1040-NR, you need to use another tax preparer, and that includes 2021, because on an F-1 visa you file form 1040-NR even if you are living in the US (unless you have been here more than 5 years). And if you used Turbotax for 2020 when you should have filed a 1040-NR, you will need to file an amended return as a 1040-NR. You may want professional help.
Claiming the IRS 1341 claim of right tax credit on your 2022 form 1040-NR may require manual entries that can't be done by your software provider, you may need to file on your own to make the claim.
Some other important comments.
1. Your IRC 1341 credit would only return your federal income tax. If you paid state income tax on that money, you will have to check with the state tax department for the proper procedure.
2. Your claim of right will not return any social security tax. You should be exempt from SS tax as an F-1, as long as the employment is closely connected with the academic purpose of the F-1 visa. If the employment is not connected with the academic purpose of the F-1 visa, you should have had social security tax withheld. To get a refund of the social security tax on the repaid wages, you need to file form 843. You would have to explain the reason for the refund, include proof of the previous withholding (W-2) and proof of the wage repayment (contract, emails, canceled check) and include a letter from the employer verifying that they will not be refunding the social security tax themselves. Form 843 is not included in Turbotax and is not part of your year-end tax return, you can file it as soon as you have the documentation needed to make the claim.
3. Finally, an important word about the IRC 1341 Claim of Right Credit.
You can only claim this credit if you had a reasonable belief at the time you accepted the bonus that you had an unrestricted right to the bonus. The IRS might argue that since you signed a contingent contract, you knew that your right to the money was not unrestricted. Or, the IRS might argue that if you were planning all along to return home after 2 years, you knew or should have known that you did not have an unrestricted right to the money. I'm not saying you should not try to claim the credit, but I'm saying that the IRS might attempt to deny the credit if they argue you did not have a reasonable expectation of an unrestricted right to the money.
If you did not have a reasonable expectation of an unrestricted right to the money, you can't claim the refundable tax credit under IRC1341, but you can still claim the repayment as a special itemized deduction not subject to the 2% rule. The problem in your case, is that an itemized deduction will not benefit you on your 2022 return.
Best of luck.
