Well, Living History is about being as close to accurate as possible, so when a group that has different goals tries to participate, I can see where the conflict is. Neither group is "wrong"; there's simply a different emphasis.
I just don't like to see inaccuracies depicted as historical truth.
Most groups outside of Living History, including the SCA, will overlook a lot as long as it has a "medieval look". But there will always be accuracy-nazis to be found.
Anyway, since these guys are relatively local, it's always possible we'll run into some of them at an event or something. Might have to keep our eyes open.