The ethical dilemma is not on your end; the ethical issue is what should the company do.
They set a deadline, and now they are considering allowing an exception to that deadline.
While they do want to make sure they get qualified candidates, they don't want to run the risk that they never make a decision. If they always reopen the system for late applications they can convince themselves that next week the perfect candidate will apply.
They also risk that a candidate will claim the constantly shifting process was discriminatory.
This doesn't mean don't ask. This doesn't mean don't plead for the opportunity to enter the process after the deadline. But do realize that they may say no.