I don't want to use the dovetail system that is machined into my receiver, but a traditional base/ring system.

I prefer a dovetail front and windage adustable rear, Leupold or Redfield type system.

What base fits the Mk. XXII ?

Caliber is .17 HMR, but I don't think that makes a difference

Neither Weatherby or Leupold have a chart for matching bases to rifles on their web sites.

Any help here?

Brownells has what you want.
They are made by Redfield.
I bought a set of bases for my WBY XXII from them.