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Since a 3 handle valve is dated (forget about getting parts)and the trim unlikely to match you're better off going with all new fixtures. Not to mention a 3 handle valve is not pressure balancing (no anti scalding/cold shower protection)
I agree on that. Is there another way to piece together separate units, or again does it make sense to simply buy a set as to avoid additional labor costs from the in wall plumbing work?
Every shower fixture manufacturer has valve, hand shower and diverter components that can be assembled into a matching set. You aren't limited to a pre-packaged set but that does simplify the choices.
I have to step away for a few minutes. If you don't have any more questions I'd appreciate a favorable rating. If you have more questions I respond when I get back