I am kind of in a situation like yours. I am multi culti myself (Dutch Suriname/American) and as well as that I've lived pretty much all over. My kids are also half Serbian, and live in Serbia for the moment, although that might change later on. As a single mom, I am able to speak Dutch to the kids while at home, and Serbian outside of the house. I'd also like them to learn English, as well as ideally some of the other languages I picked up on the way. I do find that, as a woman on my own, there is only so much I can physically achieve! Besides, I have to say that DD mainly speaks Serbian, although she can understand Dutch perfectly she won't speak it as much and answers in Serbian most of the time, even when spoken to in Dutch. Adding yet another language will get too confusing right now, I feel.
Is it possible for your husband's family to be involved with the process of your DD learning French and understanding the culture? I speak with my family through Skype a lot, which is free! That way my DD picks up a *little* bit of English, as well as be able to see people (through webcam) who she would not know otherwise. I think that is great!
I'm Olivia. I blog about physiological childbirth, homebirth, and unassisted homebirth!