I don't think 5 is too young to be sitting at a desk. However the EYFS is about learning by play. I appreciate your DS will need facilitation, but if the desk itself is an issue, why can't they start by doing an activity on the floor?
If the issue is 'attending to the demands of an adult', then it's different. But don't let them call that 'won't sit at a desk' 
I missed that SS was your preference. In case you didn't know, Special Schools don't have the same restrictions as MS for numbers . They have a 'Global capacity' - in other words, the school can take up to, say, 140 children, but there are no restrictions on class size, other than the health, safety and well-being of the children in each class.
Special schools never have class sizes big enough to trigger Infant Class Size regulations, anyway.
People can and do appeal to the First Tier Tribunal every year for a special school place when they've been turned down, and as strenuously as the school says it can't cope with another child, it is up to the tribunal as to whether the child gets the place...
I think your action plan is:
-Ask for a meeting with the professionals and teaching staff, to come up with an action plan for behaviour.
-Ask whether sitting at a desk is on his IEP. If it is, what are the steps to build up to that? If his issue is adult attention, they may well need to start with a reward simply for looking at the TA when she calls his name, or giving her something when she asks.
-Ask for outreach from SS, if they provide it. (I'd be tempted to phone the SS and ask for advice, even if you get 'go through your school...')
-Offer to lodge an appeal with the FTT to get a place in SS (even if practically, you don't get a date for a few months, the LA will have to respond).
-Log every incident.
Practical advice for TA (she doesn't sound very experienced!) - Tubigrip on arms if at risk of being pinched/scratched, long sleeves, preferably t-shirt then jumper.
If she's being scratched/pinched, she must be in close contact with your DS at that time. Check whether physical force is being used to make him comply??