"Exempt" means that you are exempt from the onerous requirement of receiving most of the protections within the law (including overtime). If you don't like that, you can get a non-exempt job (probably at a reduced salary and significantly reduced stature). Otherwise, just shut up.
I've spent my whole career watching people try to "get their rights" while in the same company with the same supervisor where I felt well treated. I always took the time I needed to take. The difference was I got my work done, if something needed to be done, I did it.
When I've had employees who demanded comp time for travel, I stopped sending them out of town. When they started an "I need to go to the doctor" discussion with "I've been working a bunch of overtime and need to take an hour to go to the dentist, is that OK?" I always got really concerned and started watching their hours--typically they were the last to arrive in the morning and the first to leave in the evening and the only folks to take a regular break. When someone came to me and said "gotta go to the doctor, be back when I get here", I knew their conscience was clear and I forgot about it.
If someone brought me a Dept of Labor guide book to show me they had the right to comp time or time off to take a kid to the doctor I immediately started doing two things: (1) complying absolutely with the letter of every word in the DOL regulations (there is some ugly stuff in there to protect both the employee and the employer); and (2) looking for a way to get rid of the barracks lawyer. This only happened one time and the individual was transferred to another group within a month (the fact that she was fired from the other group 3 months later had nothing to do with me, I wasn't asked for a recommendation).
You've chosen to be a Mechanical Engineer. With that choice comes pretty good money, challenging work assignments, and usually pretty good working conditions. It also comes with 60 hour weeks when the work demands it and if you feel you must keep track of your overtime you are in the wrong line of work.
David