Replacing dry white wine with bubbly
I would not mind using a Cava in the Brut range.  To me that would be like substituting hints of pears with hints of green apples. French onion soup can be made with different stocks, so I am assuming you are using a chicken stock recipe, which typically calls for a dry white wine.  If you are using a beef or veal stock, a low sugar Lambrusco may work, but I can't imagine the flavor.

I use the wine while frying the onions (and not just in the recipe :-).  The alcohol helps creates new flavors and it also dissolves some extra ingredients. I find the resulting flavor is richer.  As the onions are just as important as the broth, I like the technique Kenji López-Alt's developed after frying 20 lbs of onions.  Dashes of wine add sugar and water as he suggests (one can skip the baking soda part for the soup).  

With the French onion soup I like Pinot Gris, which reminds me of pears, green tea, and lemons and is not too sweet, with under 12 grams of sugar per liter.  I find that the wine flavor does not blend if it is too sweet.  The flavors of a Cava are more green apples and spices and their sweetness levels can vary dramatically.  They go Brut Nature, Extra Brut, Brut, and Extra Dry, with the extra dry being the sweetest with 12 to 17 grams of sugar per liter.