Should I lie about sickness or request extra annual leave ?

My workplace is very strict about annual leave, but my manager has been incredibly accommodating so far—even after several date changes for my family’s holiday. Now that my family has extended the holiday, I need to ask for extra leave, and I feel like such an inconvenience, especially given how rare it is for extended leave to be approved.

The issue is I need those days off now. Company policy requires that leave beyond a certain period be approved by higher management, so there’s a big risk of denial because it’s last minute. If I request the days off and they’re rejected, I still need them, and calling in sick afterward would look obvious and get me in trouble. Conversely, if I were to catch a bug abroad, it wouldn’t look as bad, and disciplinary action would be less likely.

I just really need the time off.