I am on a compressed schedule and I work four 10-hour days per week. What if there is a event/meeting I do not want to miss and it is on my scheduled day off? Can I switch to another day off for that week without going through the FWA approval process?

This is what we would refer to as a “one-off” situation. You would not go through the FWA approval process for this sort of short term/one-off situation. Flexible work arrangements are for schedule changes anticipated to be longer term, two weeks or more in duration. You should be working out short-term requests at the department level, coordinating with your supervisor.