IIRC the Burris rings don't want to get tight on weaver bases? Maybe they don't want to get tight on the picatinny rails... I forget.
Good solutions, I was too tired to try to think about solutions last night.
An ounce or so ain't gonna kill me.
And I'm not averse to a one piece base either. But not really needed
I have some of the Burris Signature Zee rings installed on Weaver bases with no issues. I did get a set of used ones once that had been installed on a wider base of some sort, they did not snug down enough to suit me so I put them into a vise and gently squeezed the clamping portion at the base of the rings a little closer together until they were just a bit snug when sliding them on. I have never had that issue with new ones.
