Definitely go with Tikka, they make a perfect ACTION, stellar factory barrel (cold forged and polished) and have abundant after market options
I have a Tikka T3X Lite in .30-06, 7PRC (Custom Carbon Barrel), and 6.5 PRC.....
6.5PRC is currently the only caliber they are offering with a fast enough twist rate for running 'heavy for caliber' high BC bullets in a magnum caliber. The problem with the 6.5 PRC is the 24.3" factory barrel, and you dont need the extended range, so its really not optimal for your scenario.
What other caliber options are you finding for the short barrel tikka?
I'd look for the thinnest possible bullet diameter down to 6.5mm, run the heaviest hunting bullet in the caliber you choose, to optimize your sectional density. For what you are doing out to 350Y, .308 should be fine, just a good idea to double check your impact velocity and make sure (if you are shooting copper bullets for instance) that they will sufficiently expand etc....The short barrel is going to bring your velocities down by about 30FPS per inch of barrel length