Based on your question, you do not have a 2023 form 1099-R. Which is odd because the form is generated when the RMD is processed out. Since you received the check on Jan 2, it seems unlikely that it was processed in 2024.
You used the word check rather than transfer of funds. A transfer could have been done in 2024 but a check mailed would not be reasonable for 2024.
I would file both RMD forms for 2024. If the IRS asks about 2023, you can show them it was received Jan 2 which means you clearly requested it in 2023. You will want to keep any paperwork proving that you did take the RMD in 2023 - despite when it was received.
You can file with 2023 if you want, you just don't have the IRS with a matching form.
what year is on the 1099-R if it's 2023 you are in the clear. you are deemed to have received the RMD in 2023. placed in the mail rule.
The date that the check is shown on their books as paid to you is the date that the check is placed in the mail. So they issue their tax forms in accordance with that.