Since you said PushEvent API I can infer that you are using Campaign Classic.
This appears to be an invalid use case.
MC is designed to be used for transactional messages i.e. password reset email, temp password email, sales invoice etc.
They are triggered as part of an event.
If your requirement is to limit it to 5 emails per customers, these are too many emails and work against the transactional structure itself. I'd suggest you to rethink it.
Additionally, MC works on event data and is not linked to profile database. So, transactional emails have no way to determine if the current ctx data belongs to a customer who has already received an email.
Additionally pressure rules can only be applied to a delivery which has a contact date set and enters Target ready state. Both are not true in case of a transactional email.
Based on above comments, the answer to your question is that it is not possible and in an ideal scenario there should be no reason to apply a limit on transactional emails.