Considering the weather, what is the best time to go and considering the cost of airfares, what might be the best time to consider?
These tours only run from May 25 through the end of September to help moderate possible weather issues. "Summer" airfares tend to apply to the whole range of dates, but may be a bit lower on the first and last dates (before and after the typical school vacations), the same dates that may have more weather risk. I'll let someone else address late May versus September specifically.
September is best. May and June have snow on some of the good trails in the alps.