My lightweight Weatherby has given me problems with mounts where I have had to shim (3mm) the front mount for the scope to work. I found Leupold Bases for the LW Weatherby fit without the need.
I prefer Burris rings, can I use Burris rings in Leupold bases??

Yes. Rings are pretty well standardized. You just need to make sure that you get the rings that match your base style. If you have Leupold bases, then there's a 99% change you're going to need a set of "standard" rings. A turn-in, dovetail front, with a windage adjustable rear.

Ditto what Kingfisher said. The standard "turn-in" front ring with the grooved rear ring is commonly known as the Redfield-style mount. The Burris, Leupold, and Millett "turn-in" front dovetail rings are interchageable with all standard turn-in bases, including Redfield.

Caution: Do not get DOUBLE DOVETAIL rings. On this type ring both the front and rear rings are turn-in.

If your old bases were off by 3mm, then they were mismatched. The actual thickness of the bases varies from manufacturer to manufacturer, so you MAY have had a rear base from one manufacturer and a front base from another. Manufacturers also may change the thickness of a base over time, thus you COULD have had bases from the same manufacturer but mismatched as to lot.