Go outside the tourist season. Summers are horrible in Paris anyway, since it tends to get quite hot and humid, and public transport only barely starts having AC.
You may get cold and rain in the winter months, but those can be worked around: a sweater and an umbrella will do the job. When it's 40ºC outside and 60% humidity, there's nothing you can do.
Winter definitely sounds like the way to go, thanks for the idea. I've heard about the awful summers but never even considered going in winter for some reason.
I can second visiting Paris in February. Instead of 40+ degrees and massive hoards of tourists. You'll have 10 degrees, rain, and merely large hoards of tourists.
You may get cold and rain in the winter months, but those can be worked around: a sweater and an umbrella will do the job. When it's 40ºC outside and 60% humidity, there's nothing you can do.