I presume you're going to leave the Weaver base on the 20 guage barrel and get a new base for the .308 barrel. In that case, all you need is Weaver style rings which will interchange with the bases. I believe, however, that the bases for the two barrels are the same. Get a dial indicater and check the diameter of the respective barrels where the scope goes and then you'll be sure.
The gap between the barrel and the front of Weaver bases is normal. If you don't want the gap, get Duo Rings from Thompson/Center.
I'm going more toward Leupold scopes especially with heavy recoiling calibers. I would say, try the Bushnell Banner scope and if it breaks, try something else.