Plot contrivance aside, how is it in Dawnguard that the vampire curse can so easily "burn away" the werewolf curse?

The same reason the Volkihar now look like Underworld vampires: plot.

Seriously, I never got that either.

Oddly enough, if you're a vampire when you go through the become-a-werewolf part of the Companions questline, your vampirism will be cured.

Maybe you could headcanon yourself into thinking that the symptoms are "suppressed" somehow - in Morrowind, the divine disease Corprus could not be cured but Divayth Fyr gave you an experimental potion that was capable of removing the physical effects of it apart from disease resistance (which, again for plot reasons, did not extend to the scripted forcing of werewolf disease in the Bloodmoon main quest). Perhaps the Volkihar have the magickal know-how to stop the effects of Lycanthropy, although in game it appears as a complete removal, and similarly the curse of Hircine removes the benefits and downsides of vampirism while the state itself is technically not so easily removed? However, this doesn't really present any different a problem - lycanthropy being Hircine's influence on Mundus, I doubt that he would be any more pleased about the tampering with its effects than an outright cure.