I believe this only applies to movements within Dubai/Emirates... I am pretty sure it is not applicable to people moving here from other countries... as such... a move at the beginning of T3 is an ideal time as you are not competing with those already in the system wanting places at different schools.
As far as I am aware there is no KHDA relating to new pupils arriving in Dubai and being registered to start their schooling here for the first time. There are rules about changing schools [b'>within[/b'> Dubai and deadline dates for making inter Dubai school moves.
For new arrivals, admitting pupils is dependant on school admission policy. I don't know of a school which doesn't admit in Term 3, providing they have a place.
Good luck with your search, I am sure you will be able to find a good solution for your children - which won't entail missing 5 months of their education.
I have known many kids who have been allowed to start in schools after the Spring/Easter break - I am not sure what the term dates are for this year but I know Easter is very late. The main problem is finding a school place, and I believe you'll find this a lot easier in March/April than in September.
Personally I would try to find places for your kids to start at the beginning of the last term, last year I know of families who found school places in Dubai British School, Kings and Star Int. to start their kids then.