Congratulations to you and your dh on the birth of your baby daughter!
It sometimes depends on the child, but speaking generally, three languages no problem. Particularly if you start from birth as you are doing.
We are an English family living in Belgium. We speak English at home. DD is being taught French and Flemish at school. She has had no problem picking up the three languages (she started at 2.5 yrs and is now 8 yrs). She switches easily between the three.
I'm no expert but I'd be careful with the Greenglish as the school have emphasised how important it is to eliminate any confusion or muddle between languages. DD is only taught specific languages by specific teachers and that never changes. There is no mixing of languages, ever. The two languages are separated very carefully.
So one parent/one language - sounds like an excellent idea but you have to be rigorous about it. If you don't think it is going to be practical, then I wouldn't try.
It might be worth finding out whether your child is going to be taught English at her French school, and if so, at what age? If she is then you could just opt to support her at the time when that happens or (if you want to have an early start in English - there is lots of evidence to suggest the earlier the better) then you could start her off with English lessons once or twice a week (or you could choose one or two days a week when you all speak English - although I don't know if that would be classed as 'muddling').
Good luck with it!! Your dd will have a great start in life. There is lots of evidence available now to suggest that babies who learn more than one language are better at problem solving and think strategically (something to do with the way their neural pathways develop I think !!)