Telling us you already have a tikka t3 lite in 300 WSM
& a Rem 700 in an HS precision stock in 7mm Rem mag just changed the situation GREATLY! Since you already have a .300 WSM & 7mm Rem mag, you already have appropriate rifles for the larger game & we are really looking at what's appropriate for the smaller game on your list.
Now with the choice between a .260 Rem & a 6.5-06 it's a no-brainer, go with the .260 Rem (or possibly a .25-06). Each of these has low recoil, readily available factory ammunition if needed, and each is known to be inherently accurate. Each will take you to the range/game size where you need your larger guns while being comfortable shooting more rounds than you would want to shoot from the magnums.
The tikka t3 lite you own is a great cary gun. The problem with the Tikka is that it's optimized for being light, which makes it great to cary but less than fun to shoot with that powerful of a round. The Tikkas tend to be very accurate factory rifles. Putting a heavier barrel on the Tikka and/or a muzzle brake
would tame the recoil & you may then find you already own the long-range elk rifle you want.