What about the Al Mawakeb school
AMS Dubai

There are a lot of Arab students there, and the languages taught are Arabic, French and English.

My neighbours children attend there and are very happy with the school.
There is a branch in Al Barsha ( near the Mall of the Emirates), and another one in Al Garhoud (which is near the airport)

The French schools I have heard of are the one you mentioned (AFLEC- Lycee Francais International, and LGP (Lycee Georges Pompidou.

It is really up to the school as to whether they accept late places..you may need to ring and find out. (as schools have now finished for 8-10 weeks, best you try them now, as some may have skeleton staff working for next few days/week)