I honestly hope Alizee never makes an album to specifically target the huge English-speaking market. I know some international artists sing in English as opposed to their native language for this purpose, but it would be a mistake unless those songs are originally English songs. For example, Alizee's cover version of La Isla Bonita is perfect, but the English version of Jen ai' Marre is butchered to appeal to an English-speaking audience.
I remember Alizee addressing the subject of singing in English. She said she was trying to learn English because it was ridiculous singing in a language she didn't even understand because she couldn't put her emotions and inflection in words she didn't know. I agree. So Alizee, keep singing in French. We love your voice even though we don't know what you're saying.
As for songs with French lyrics succeeding in the English-speaking market, the only non-English song (much less French) that I can even think of was Nena's German 99 Luftballoons (which they later butchered to make an English version). I'm pretty sure Alizee would have succeeded in the U.S. with Jen ai' Marre, Moi Lolita, and a few other songs in their original French versions.
Blondie sang their hit Sunday Girl in French. I wonder if this song sounds butchered in French:
I like both English and French versions.