The tub should be supported underneath where it sits on the subfloor, not just at the lip. Usually the manufacturer specifies that the tub be set into a bed of thinset mortar to make sure it's well supported across the entire bottom surface. We have a tub that's doing the same thing.
The easy fix is to remove all the caulk around the edges, then fill the tub with water before recaulking, only draining the tub after the caulk has had time to dry fully. This ensures that the caulk seam stays mostly in compression even when the tub is full of water and people, reducing the chance that it will get pulled apart. I've done this to ours and it's lasted well so far (granted, it's only been a few months now -- we just moved here in April).
What I'd eventually like to do, as our fiberglass tub still creaks when you walk in it, is to open up the adjacent wall and try to get thinset mortar underneath the tub. My plan is to gently pry the base of the tub upward with shims, then slather mortar underneath as much as possible, and then remove the shims so that the tub settles into the mortar. This might be an option for you, but it might involve some tile work depending on where you can get access to the subfloor under the tub.
I know this doesn't do much to answer your specific question about the feasibility of removing and replacing your tub. That's difficult to answer without knowing exactly how your tub is installed. I have to think that's going to be a big job though, no matter how it's installed.