JavaDad, that is absolutely not the case.
The reason the school is unlikely to want you to apply for a Statement is that they would then have to meet the needs of your DS and that money would have to come out of their budget for the first £6k.
Imagine this:
You are given £12,000 to spend on whatever you think you need at home. Some of it is intended to go towards decorating, but you decide that you can live with the house as it is. You have £12,000 to spend.
OR
You have £12,000 to spend on whatever you need at home. However, £6000 is given to you in decorating vouchers, so you have to spend it on decoration. You then have £6000 to spend on everything else. You wish they hadn't done that, because actually you wouldn't have chosen to decorate so nicely and would prefer to spend more on a fitted kitchen.
That's the reality of SEN funding. Schools are given a 'notional budget' for SEN. It isn't ringfenced - that means that they can spend it on anything. If a child has a Statement, they have to meet the child's needs as specified in the statement, which means that they have less money to spend on flower borders things that benefit the whole school.
In your situation, especially if you think Special School is a realistic necessity, you should write to your LA's Chief SEN Officer without delay, requesting a Statutory Assessment. If it is agreed, you'll see an Ed Psych anyway, so there is absolutely no sense in the HT telling you that 'Ed Psych will be more helpful'. It's not an 'either or' situation.