It may not be legitimate old myth, but I'm certain the idea that there's enmity between vampires and werewolves predates Underworld. Underworld likely got it from World of Darkness, but it definitely predates that, as well. I remember an episode of The Real Ghostbusters that relied on this idea (in 1987).
The thing that a lot of people in this thread seem to be forgetting is that the Companions aren't historically werewolves. This matter of the "beast blood" has been with them for hundreds of years, but the Companions have existed for thousands. The Dawnguard would put those Companions who aren't pro-werewolf (including their Harbinger, assuming you went this route) in an awkward position. On the one hand, they have no love for the curse placed on them by the Glenmoril Coven and would likely prefer to consider the Dawnguard allies, or at least not be at odds with them.
The problem is Aela. She does enjoy the beast blood, and though she will likely not be permitted to induct any new Companions into it, Nord honor would require that, so long as she remained a member in good standing, they stand by her as she stands by them. Absent the Dawnguard, they would likely consider her choice to be purely a matter of personal preference. But with the Dawnguard around, she becomes a liability. So long as the Dawnguard remain strategically focused on Vampires, the two groups would likely enjoy a chilly relationship. Iskur, though he would have no love for Vampires, would likely give Aela a pass since going after her would trigger war with the Companions, which would be a terrible distraction from their core mission.
But once the Vampire threat is dealt with, a confrontation would occur, and the Dawnguard would likely insist that Aela either get cured or be turned over to them. The Compaions response would depend on the behavior of their Harbinger. If he wavered, the controversy might even tear the Companions apart from within, with some siding with Aela, others demanding that she get cured. The damage could be mitigated somewhat if the Harbinger decided to demand that Aela get cured, himself... but it would likely damage internal relations for some time, if not force a formal split between "pro-beast" and "anti-beast" wings of the Companions. Only if the Harbinger adopted a strong "She's one of us, and we will stand by her, no matter what the Dawnguard say" would the Companions remain solid.
At that point, it would come down to the decisions of two individuals: Iskur and Aela. Is Iskur prepared to go to all-out war against one of Skyrim's most beloved institutions over a single werewolf? Is Aela prepared to drag the Companions into a potentially unnecessary fight for the sake of her tie to Hircine? Iskur might back down; Aela might decide to get cured on her own, out of respect for her comrades who are prepared to fight and die for her. Or... the war would occur. I suspect the Dawnguard would be the ones to lose, mainly because, without the Vampire threat dealt with, they don't have as much reason to exist. It's possible some of the membership would split off and join/form a newly formed chapter of the Vigil, which might make an exception for Aela provided she behaves herself.