Hi there.
What you have been told by the school is utter rubbish - very irresponsible of them to feed you this garbage although, sadly, not uncommon.
It is not the school's decision as to whether to issue a statement for your child - it is the LEA's. And yes, LEAs are almost always unwilling (initially) to issue statements because it costs them money. The funding for a statemented child - for example, the funding for a full time 1:1 TA for your child - would no longer come out of the school's budget, but out of the LEA's. And any provision detailed in a statement is legally binding (which any support at SA/SA+ is not). Which means it becomes a legal right of your child to receive it, and a legal obligation on the part of the LEA to provide it. So - expensive for the lEA and makes them more accountable for your child. Of course they don't like statements!
However, they are regularly issued to childrenw ith SEN, although often (mostly?) after a fight.
You have the right, as a parent to request a statutory assessment of your child's needs. See the IPSEA website for a model letter, and ring them for advice - they're hard to get hold of, but very good.
You do NOT need the school's approval or support to do this, although it might be a good idea to tell them (in writing, as well as verbally) that you are intending to do so. If there is any way they can come around to supporting your request it would be a good thing.
If your request for SA is turned down, you have the right to appeal.
Good luck and come on to the SN boards for loads of great advice and support.