I'm a medium D cup and while I can work out comfortably in a binder(or even two, at times) I've gotten away with using two sports bras in sizes much smaller than I would normally wear, one backwards, the other on normally and a loose shirt. It brings me down to a small B, large A, and the shirt helps hide that too.
Also, do you "pass" as male in public? Do people call you sir without being prompted as such? If so, I think you might want to try running into the men's room without your dad at the gym and scope it out. If there are stalls, then you might be able to get away with doing your business there.
I've gone into men's bathrooms and locker rooms without being looked at odd or hurt or..ergh...raped. But the times I've gone in, I've been consistently passing as male. YMMV, of course, just my two cents.