Flights are indeed a problem - I'm pretty sure I've heard something about it recently, there are some limits that the Israeli government imposes on the airlines and the situation with flights to France seems to be especially problematic.
Regarding your staying in Israel - did you consider a hostel? There is Momo's hostel, for example, which offers anything from single rooms to 8-bed rooms, with prices starting at 58 NIS a night in a dormitory room (for some reason they only declare the prices in the Hebrew part of the site). If you're OK with one room for the whole family, it could be a solution for you. You can find more hostels in Israel in this site, and as far as I know their prices are more or less the same as in Momo's.
There was an article about the Tel Aviv hostels in a local newspapr here just a week or two ago, and they said they are expecting to be full for most of the summer, so I suggest you act early in case you're interested in this option