Not heavy, bulky and unwieldy. I have been hunting off and on with SRHs for a long time and never found them too heavy. I think with an optic, a 7.5-inch SRH on the belt would be rather bulky (for lack of a better term).
To the OP, a good chest rig is the least obtrusive set up I think I have ever used. They are actually quite ergonomic.