@Clevinger: I think you have it pegged pretty well. The .243 will not be as powerful for big game, while the .308 will have significant recoil in a light rifle.
Since you intend to use this rifle as a dual purpose gun (small to big game), I would choose the .243. The .243 will provide more flexibility and should be powerful enough for deer. However, I would not recommend the .243 for elk.
Just to throw something else in the mix, you may want to consider the .25-'06, which is also chambered in the Tikka. If I recall correctly the Tikka only comes in a long action, and so there is no overall length advantage from the shorter cartridges. The .25-'06 would provide more power and the option of heavier bullets versus the 6mm's, if you're concerned about that.
Edit: I just looked up the specifications on the .25-'06 tikka. It has a 22.4 inch barrel. The .25-'06 muzzle blast is very noticeable from a 24 inch barrel, and so a 22 inch barrel would be even worse. Plus , there would be velocity loss and you might not wind up any better than a .243.
What I would really recommend is this: Fortunately in this country (for now at least) you can own more than one gun. Go buy yourself a dedicated deer rifle chambered for a deer cartridge. Then go buy yourself a bolt action .223 for the predators and small game. They're plentiful and cheap. Or even better, buy yourself a .22 hornet. It's quiet, no recoil, cheap to shoot (if you reload), and powerful enough for varmints up to and including coyotes. The downside is that it's about a 150 yard cartridge but I rarely shoot farther than that.