Also a 22" barrel is great but 300 WM or in fact in any belted magnum cartridge, you are really losing velocity when you go below 24", to the point that you might be better off with a non magnum caliber. Out of that 22" tube you are blowing a lot of unburned powder out the front in 300 WM.
Most folks as a rule of thumb stick with 24" ish in any belted magnum. It is just a lot of powder to burn in a shorter barrel. When you start trimming below that too much, a 300 win mag turns into a very loud, hard-kicking, expensive to shoot, 30-06. The larger bore calibers i.e. 375 H&H and 338 Win Mag are more forgiving to short barrels because of the bore diameter.
Also as you already are saying, you will need a suppressor or a muzzle brake! Even with a heavy 20 ounce scope, bringing the weight to about 7 1/4 lbs, my Tikka 300 Win mag super light kicked the same or worse than my 375 H&H.
My Tikka now wears a 7" suppressor and with the can in place (bringing total weight to just over 8 lbs) it's about like shooting a 30-06 with 180gr loads.
Just as a data point, that is the 24" factory super light barrel, now with a 7" titanium suppressor hanging off it...zero loss in accuracy, it was a half minute rifle from the factory and it's a half minute rifle with the can hanging off it.
Again...I am thinking you will save a lot of money just buying a factory Tikka super light and I doubt you will give up any practical accuracy at all (maybe like .15 MOA or less) and also, you will always get your money back out of a factory rifle and *never* get your money back out of a custom rifle you've built off a factory action. Again, coming from a guy who has a lot of custom rifles.
That said, maybe the fun is in building this rifle. Dumb question but the action holes in the blank Brown stock will work with the Tikka? Or are you looking at bedding pillars in the right spot? That's not hard but it's work and setup. Also it might be a bitch to get the bottom metal bedded in at the right height if you are not working from somewhat prefit stock. I bed a lot of rifles and getting the feeding right when you are not working with a good pre-fit stock is a biatch, sometimes, takes a couple bedding steps to get the height right, test the feeding, before you finally bed the bottom metal in. But again, if you love doing the work itself...
I hope I don't sound patronizing, because maybe you already know this shit. If so, my apologies. And take all of the above with a grain of salt. I've built a lot of rifles and lost a ton of money doing it and also put a ton of time into getting rifles to shoot well. So when i can buy an off the shelf gun that will do 99% of what I need...I very happily buy it and put the money into ammo and trips and the time into trips to the range or trips to the woods.
Regardless of my a-hole opinions...have fun with yer new project!!!!