From BSP into straight French (public) school might be tough. Bilingual is a good option, working towards the Britisih Section OIB which is an excellent passport to universities in the UK and France. I know that is looking many years down the line for you, but you only keep your options open by planning that far ahead sometimes!
For bilingual options the Lycée International is a great school.
Some "sous contrat" privates are much more expensive, natation, surprisingly enough, though I would agree that the figures you give are the norm. But Ermitage for example is much higher, depending on the programme you choose it can be around 13,000 in the higher years, with big hoiks each new school year, and so many unannounced or glossed-over "additional extras" (that are hardly optional) to keep up the financial squeeze.
Bear in mind bilingual syllabus puts extra demands on the kids and that transition from there back to straight French is difficult (something has to give and it tends to be French history, sciences and literature in the bilingual schools).
I would be less swayed by the consideration of whether your children's classmates will be staying for the long term or not, when it comes to international or bilingual schools. Many families are there for the long run, and even if your best buddy moves on, it can be quite positive (possibility to keep in touch via FB/email, or even get together during school holidays in their native country/new posting -- my kids have benefited culturally from this scenario and improved their third language to boot).
Naturally we can't give you the answer though! Just some things to consider...