Beartic is really relying on the Move Tutors to give him something to give him a niche that Mamoswine can't take or at least make inferior.
Right now, Beartic actually does have 3 things over Mamoswine, Taunt, Encore, and Swords Dance. Unfortunately no one seems to realize that or want to use it because Taunt and even Sword Dance isn't all that good on a Pokemon that can't take more than 2 hits from anything due to it's typing. Beartic's no slouch defensively, but his typing & speed ruin any Swords Dance sweeps he would try to set up.
Beartic needs a few things. Agility would be any excellent start for the bear. 350 Attack Stat + Life Orb would be enough to pull off a late game sweep. Next thing he'd ThunderPunch just to better deal with those Water-types. Zen Headbutt for Fighting-types. (though they'd KO him with Mach Punch anyway)
But aside from Fire Punch...yeah...there's nothing he could realistically get from tutors that would be meaningful...Ice Shard or even Aqua Jet maybe, but that's kind of it. Most of the moves he would love like Waterfall & Wild Charge are already TMs he can't learn.
Poor Snowbear.