Bear in mind, I've NEVER touched a Savage action.
Be that as it may, here's my two cents.
I'm guessing the Savage is an 1.350" diameter receiver. IF this is correct, it puts right on par with a majority of other mainstream and custom actions out there.
I've built 338 Lapua Magnums in an Ackley Improved configuration on Remmy 700's. The rounds were loaded to brutal pressures and the receiver never complained once.
As we know, the Remington is a 1.625X16tpi receiver. Don't know what the Savage is, guessing a 1.0625X18tpi which if correct, is the same as a Nesika. I've build close to a hundred 338LM's on Nesika actions so there's no worries there.
The bolt will certainly have to be opened up. We (meaning Nesika) use .590" for a BF diameter. It's worked without issue, so maybe that'll help some.
Ejection ports always seem to be the pain in the nutz when it comes to these things. I'd avoid removing material off the front end of the port if at all possible since that's where all the business takes place. If something can be trimmed off the back though without goofing with scope mounts or other peripheral feature, then blaze away on that milling machine and make it happen. Nothing wrong with that IMO.
If the action in present form will tolerate the 300RUM, I'd have to speculate it'll run just fine with the Lapua. The Remmy I did was a 30-06 when I started. My buddy that owns the gun isn't known for being very conservative with a powder scale. He likes speed and with an AI configuration it gets it done pretty well with the little 225 grain bullets. (and a 36" barrel)
Good luck, post some pics if you can.
Cheers,
~NC