To be honest I'm really not surprised about that, it's fairly typical in my experience for the kind of wishy-washy positions generally taken by the Muslim organizations in Western countries, which seek some impossible balance between being sufficiently politically correct to not be branded extremists by the political mainstream, while still satisfying even the more rebellious elements among the people they represent. They often fail in both respects, sometimes even at the same time, but one can see why they keep trying. CAIR is not very different in that sense.
The sad part is that "Yusuf Al Qaradawi enjoys unparalleled respect and influence throughout the Muslim world" is really close enough to the truth to make it entirely justifiable as a statement by an organization as impossibly placed as the MCB (or CAIR). The man is an idiot, but an influential one. Still, I don't think releasing such statements or sponsoring him for a visum is irreconcileable with taking a pro-peace and anti-violence stance, as such. If you're going to limit your selection of invited speakers or guests to those Arab Muslims who publicly and invariably condemn Hamas and Hizbullah and defend Israel, well, your list is going to be very, very short and you're making yourself largely irrelevant at least among the people you're supposed to represent (even if you become a darling of some other groups in the process). That doesn't have that much to do with religion and all the more with nationalism and politics. If you think political correctness in the west is stifling debate, you've clearly never taken a look at the Arab countries - except that their brand of political correctness has some very different dogmas and truths that must never be spoken than ours.