I’m natively bilingual (Arabic & English), even though both my parents are natively monolingual (Arabic; although they speak English quite well). I went to an international British school from my first year of education up to my final year, which is to say from KG1 all the way to the sixth form. My school had a strict English-only rule, and we weren’t allowed to use any other language except in Arabic and Islamic Studies classes. Arabic was essentially treated as a second language. I also grew up consuming English-language media (mostly American). I actually had more exposure to and immersion in American culture than my local culture. I had to put a lot of effort strengthening my Arabic language and my connection to my culture after graduating.
If you want your daughter to speak French fluently from a young age, you will have to invest in it. It’s not just about the immersion, but also about how she relates to the language. I never “learnt” English. I never took English lessons and practiced English, etc., the way non-native speakers need to practice, the way a person needs to practice a musical instrument, etc. The approach of lessons and practicing makes the language feel like work and can easily be abandoned if the child finds it boring—and the truth is, the child is likely to find it boring. Your daughter needs to live and breathe French alongside English.