Fascinating question, so let's make it more accurate. Suppose the US knew about the camp but didn't tell the UK government, but just went ahead and took it out (along with any civilians nearby - hey, they were around, they were probably in league with the bad guys.) Well, knowing our government, I've no doubt at all that they would tug their forelocks and say "Thank you kindly, Master." But I'm not sure the general public would be so universally grateful.
Back in the days of the Troubles, the British military intelligence knew the IRA had bases in Eire, and in some cases probably had the map references and a list of the people there. But having an old fashioned respect for international law, we didn't send a plane over to bomb them. The interesting question is, if we had, do you think it would have weakened the IRA (whose entire political platform, remember, was that they were fighting the Brits on behalf of all the Irish people,) or improved co-operation with the Irish government?
If only it were that simple.Really, it's a question of whether we as a country identify with the terrorists, or those fighting against them. I think the fact the Pakistanis are angry about American troops fighting with the terrorists in Pakistan, rather than with the terrorists themselves, speaks volumes about their allegiance.
If someone invaded my country to attack someone they (not I) had a problem with, I would feel justified in protesting even if I had not the slightest sympathy with their targets. Wouldn't you?