1/2" Feed, 3/4" Piping, Effect on Water Flow Rate
The hot/cold mixer valve and/or the 1/2" valve outlet make installation of 3/4" pipe, downstream on the run to the shower head, unnecessary; you will have no appreciable benefit from this. Also, I imagine that your "rain shower" unit has a 1/2" inlet (although you don't mention this).

Your concern about "wasting hot water" doesn't make sense to me; your proposed pipe size increase is after the hot and cold mixer...