Holland's in the US is now offering an adjustable Picatinny rail. Sounds like an option, if you can get one. This would accept your rings.
If you want to retain the base, use Burris Signature Zee rings.
Or, go with a Redfield/Burris/Leupold style base with windage adjustment built into the rear ring mount. Burris Signatures are still worthwhile. As noted, centre the scope, use the mount to get basic windage alignment.