Question: If I have 4 damage on a hoth objective, with Desolation of Hoth, can i move ALL 4 of that damage to a unit with a damage capacity of 2? Or would i need to leave 2 damage on the hoth objective?

All four

Yeah, with Desolation of Hoth, it's designed to pretty much completely heal that objective. You assign all 4 damage at once, unlike the Protect Unit ability which is limited by the protect units health.

So, 4 damage on a unit with 2 health means that the unit is killed and all 4 damage tokens are simply removed from the board along with the killed unit.