Yes, you will file a Form 709 by the April tax filing due date, but it does not get sent in with your Form 1040NR. You can find the information on the most up-to-date version of the Form 709 here.
The annual exclusion for 2022 is $16,000 and for 2023 is $17,000. So, you will owe tax on the $3,000 in excess of the exclusion for 2023. The Basic Exclusion (or lifetime cap in your words) means that you lose any annual exclusion once your gifts have exceeded the Basic Exclusion.
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