No, but that't not all.
First, regarding the 2023 1099-NEC. The IRS accepted your return because acceptance only covers very basic facts (name, SSN, all the math is correct, all the proper boxes are filled out). It can take more than a year for the IRS to catch up to missing or mis-matched income. So the IRS could still send you a letter for the unreported income, depending on when the letter is prepared and when the corrected 1099 is processed. If you get a letter proposing a deficiency, you can respond with a letter explaining the situation and enclose copies. You don't change your 2023 tax return because what was reported was correct, and any bonus that is paid in 2024 is taxable in 2024, no matter when it was earned.
However, as regards the payment of a bonus in general, and payment of a bonus in 2024 (regardless of when it was earned):
ALL money paid to you by a W-2 employer as compensation for your work MUST be reported on your W-2 and is subject to withholding of social security, medicare, and state and federal income taxes. By paying on a 1099-NEC, the employer is breaking the law, and either accidentally or on purpose, trying to make you pay 15% self employment tax for social security and medicare instead of withholding 7.65% from you and paying a matching 7.65% themselves.
Start by pointing this out to them. Keep copies of your correspondence, especially if they refuse.
When you file your return, if they do not put the income on your W-2, there is a procedure when entering the 1099-NEC to indicate "I received this from my regular employer and it should have been on my W-2." Turbotax will prepare a corrective form so you pay the correct social security and medicare tax and not double. The employer may be investigated (if the IRS is not too busy auditing someone else.)
