I live in Zurich and things like housing costs, schools and tax vary from Kanton to Kanton, so I can't speak for Basel with any authority.
I agree re checking out Englishforum.ch, use the search function, and if you ask a question post first in 'introductions'.
The IS schools that we looked round had long waiting lists, 11 months for us, and we decided to use the local school for my dd 12. It was a hard decision to make, but so far so good, since May.
At my dd2 kindergarten there are 15 children, this makes up reception and year 1. 4 of the children have lived in England or Ireland and two more speak English as a second language. Here in Zurich there is a hell of a lot of support for integration and language aquisition.
Learning a bit of German and not following an English curriculum would not be a disaster for your dc if you find that you can't get a place at IS. Though for a two year contract I do understand why that is your first choice.
You can really take advantage of a good quality of life here, for me that would include a short a commute as possible.
If you did decide to live in France or Germany, I would try to buy somewhere using a chf mortgage so you can fix your accomadation costs. The chf is high (maybe overpriced) and if that is corrected by a strengthening Euro your rent could go up on a fx basis.
Yes food is expensive here, my weekly shop for 5 is never under 200chf, but we do eat very well and don't really try to economise. I couldn't be bothered to try to get to Germany for food shopping, but we do have friends that do.
I hope that it all pans out for you. It sounds like a great opportunity for your family.