I can't help you with your Tikka question, but suspect it would be trickier. The first problem is that Tikka only makes one length action, and it's a long action. I think short-action calibers in a Tikka action utilize a different length bolt stop to limit excessive bolt travel. This may not align with where you need the bolt to stop with a 350L round. The other oddity is that Tikka uses a three-lug bolt. This may or may not play nicely when attempting to strip a 350L round from a 223 magazine.
I would suggest taking a look at the Howa barreled actions at Brownells. They offer one in 350 Legend already. You could customize it from there with a stock of your choice. Lots of options for the Howa Mini action. I recently did this with a Howa Mini in 6.5 Grendel. I decided on a walnut Boyd's stock, and a hinged floor plate (instead of the included detachable magazine)
