Site A has Keyword X as inactive with a bid of $1 to activate. If you create another campaign and target a different site, will you still need $1 to activate or does it have the effect of reseting it? Does the history have an ongoing influence?
What I can tell for sure is that if the new landing page is more relevant for the keyword - the quality score will be higher and if the quality score is higher the Min bid will decrease from $1 - but this depend on the relevancy of the new landing page. I think that a new campaign will have a new quality score history - but I am not sure. Go on and give a try, but do not use the Copy/Move keyword tool as it will carry also the quality score history and for your case you want to reset the Quality score History