Red or White

I think either is fine. If your onions are too strong you can soak the slices in water before cooking to make them milder.

Honestly, I have done a bunch of reading about this....there are recipes using red onions for Lyonaise soup. But, its not tradional. (Apparently red onions are a premium in France so they use yellow) I have used all kinds; separately and together. I didn't find much of a difference. As some onions are stronger than others. One thing I don't use; is sweet onions; they cost more and why bother? I am caramelizing regular onions to bring out their natural sugars anyway. I say go ahead! As for the color? Its just an expectation one might have; doesn't mean it won't taste heavenly. For my clients I add a handful of spinach or Kale and a cup or so of white beans/canellini. They desire the extra nutrition and I think it makes soup more of a meal.

If you use red onions, your soup will be a funny color.

French onion would be a white (or yellow) onion.
Cook slow and low to a nice caramel color before adding stock.
Time and patience will reward you.
A splash of cognac or white vermouth does not hurt either !