I agree with Kanga completely. Even if you could afford to pay he cost of a 1-1 support, what about DBS, who will manage the 1-1, to whom will s/he be answerable during the school day and what will happen for the rest of the school day?
That's if you could find someone who could take on a role that couldn't have a contract, no job security whatsoever. If, for example, your DS was excluded from school, she'd be out of a job.
It is the school's responsibility to make arrangements to support your child and to pay the first six thousand, as has been said. They should be gathering evidence to show what works, if anything, and how he can be further supported, through advice from outside agencies and showing how acting on that advice has made a difference. Then, if things aren't improving in the way you all want, they should be applying for an EHCP, based on the detailed evidence they show, what has worked, what hasn't, advice acted upon and what additional support is required.
If he's not severe enough to warrant an EHCP, then the school has the funding within their budget to provide additional support. There is a timescale for granting an EHCP, but the LA is not going to do that without evidence of what the school has done to try to ameliorate the difficulties, what advice has been sought and the outcomes for the child of acting on that advice. Yes it does take time, butduring that time, there should be some active intervention.