Does using Reacta Clean harm/change the finish on a polished ball? 

No, but then it doesn't do much else, either.

Or should I just use the Reacta Shine for my polished equipment?

I suggest cleaning the ball after every set, and then re-polish or refresh the shine after every 15 - 25 games, more or less. It depends on how worn it gets (rev rate, lane surface both come into play here).

I'd recommend resetting the base grit level and re-polish may every 50 or so games. The surface gets worn down from use.