Entry into local schools is easy. It is just a matter of finding a school with a space - no different to most countries.
Yes, there are fewer spaces now but that is across the board and same for all. But there are always schools near you with spaces. But some, highly sought after schools either 'name brand' or 'expat friendly' are harder to get into.
And remember, that school buses will collect your child from almost anywhere on the island so distance is not such a problem.
Most
International Schools have waiting lists - especially the larger ones - so it may be just as hard to get into the
International School of your choice.
The company should sponsor you and the children for a DP - and they should have no objection to doing this because in reality it costs them next to nothing (approx $100) and the only liability is if you were to need to be deported and they would have to pay your aeroplane fare home - and that would be a very rare case.