So, basically adopting the competition / 3-gun idea? Aside from the weight and bulk, one consideration would be how readily you can switch between holding the rifle for the scope vs. the red dot. People learn to do that to be fast in a competition with different ranges to shoot at, but that's usually shooting a 5.56 sporting rifle. With the recoil of a DG rifle, even "only" a .375, it might not be good to shoot using a red dot with your head so high off the rifle. And to do that quickly under stress might not be good. I suppose it would come down to the familiarity you could achieve with the setup.
In terms of the equipment itself, I think that competition shooters usually use scopes more tailored to that purpose, and this could include the stress of mounting a red dot to the scope and then throwing the rifle into a bin before you go off and shoot some other stage. Of course the Z3 is an excellent scope but that might be pushing the intended use too far.
In my opinion, it makes more sense to keep it simple and either use something in 1-4 or 1-5 range, or in the 3-9, 2-10 or 2-12 range.
Maybe there are people on here who have done both DG hunting and the kind of competition shooting where your proposed kind of setup is used. I'm curious to see what they say.