Keep the 7mm08 and take up reloading. Buy a .22 rimfire or maybe a .22 Mag.
Or buy an '06 and load an 00 Buck over five grains of pistol powder for a squirrel/rabbit load. 110 for coyote-sized critters and 150 for deer. 180s for elk and moose.
My 700 Ti in 7mm08 weighs 6.5 pounds with scope, ammo and sling. The stock fits; it has an excellent butt-pad and is not at all bad for recoil at the bench rest. Odds are that a .308 would likely be more efficient with the 22" barrel than would the four-ounce heavier long action in '06.
You're from BATFE? Come right in! I use all your fine products!