Anything that will shoot a .300 Weatherby will handle a 6.5-300 WWH, but, like chilipeppermaniac, I'm unaware of a lever gun.
IMO, "Lever gun" and "at distance" seldom go together, regardless of chambering.
I had a long range bolt gun built in that caliber in the '90s. 30" barrel to start. Ate the throat at 700 rounds, lost 3" trying to get it recrowned far enough back and rechambered far enough forward. Got another few hundred out of it and gave up. Very flat shooting, eats 83 grains of powder per shot, but my rifle only needed 19 moa to get from a 100 yard zero to 1,000 yards.
For a hunting rifle, barrel life doesn't matter, but I'd look at a 7-300 Wby or a 7 STW for the bullet selection and bullet weight advantage. Just about all of the 6.5 heavies are match bullets.
Weatherby is the only company that chambers a rifle in that cartridge. It is available in the vanguard and mk5 bolt actions. It's a serious barrel burner and unless you handload expect to spend $4-5 a round on ammo.
There are two (at least) versions of the 6.5x300 Weatherby. The original was a wildcat 1000 yard cartridge from the late 60's(?) called the 6.5x300WWH (Weatherby Wright-Hoyer). The second is the recent reincarnation of this cartridge by Weatherby denoted the 6.5x300 Weatherby Magnum, which I do not believe is designated as "WWH". (I do not know if loading dies are interchangable between these two.) I owned a version of the first, smithed into a sporterized Mauser action with a SS match barrel free floated into a wood BR stock. Near as I could ascertain, it was made sometime during the late 1960's.
As stated above, this cartridge is a barrel burner to the extreme. As I recall, it was loaded from ~83 to 94 gr. H870 with 142 gr. SMK's. It is also an inefficient cartridge yielding little velocity advantage for its massive barrel-burning charge. If it was a great 1000 yard cartridge, it would still be used in 1000 yard competition, but it was one more wildcat experiment with little competitive future.