Without response from native speakers, I would dare to say that "archaelogical season" is for months, and "campaign" for one year or longer, or maybe shorter if it is carried out only once.
I'm not native, but I'm archaeologist and work in the UK (Oxford Archaeology, check the link in my profile). Archaeological season is the right one.
Archaeological campaign sounds more like if you were to Africa or Asia, kind of old fashioned. I hope it helps!