That sounds like a nice project. I think a rifle build is the way to go.
The COAL limitations of the rotary magazine, slow twist rates, shortish barrels, and pressure limitations of Savage's model 1899/99 lever actions make it hard to achieve a copper load that is legal for all UK deer.
However, with carte blanche as regards the action, barrel and leade, you could probably make something genuinely effective whilst retaining a classic identity.
The downside is that it would be a hand-loading-only affair, since factory loads for the .250-3000 Savage are no longer available.
A more versatile choice, albeit one with a very different vibe, would be .25 Creedmoor. For something more classic, .257 Roberts is a possibility, but takes you up to a .25-06 length action.
Apologies if I've been going over ground you've already covered here.