Yes, he would be able to receive the reward from the birthday campaign as when the campaign worker runs, it looks for users with an upcoming birthday and whether they have been rewarded via birthday campaign or not.
For example, the start date for the campaign is Sept 1, 2020, and a user has a birthday on Sept 2, the user will receive the birthday on Sept 2 when the campaign worker will run.
If further assistance is required, submit a ticket to Punchh Support. (For help submitting a ticket, click here)