Not within their remit - why not!? this is not just a school issue (if it is they do try to stay out of it), it is an issue at home too. Go to your GP and ask them to refer you for a tier 3 referral if at all possible or get your school to get the EP to refer which ever is quicker in your area.
I'm not sure this will help but when DS had a similar (though not as difficult by the sounds of it - pushing desks and chairs over but no ripping up of work or hurting anyone deliberately), the school and I worked together and put in place the following:
1 -They got emergency funding from local council for a 1:1 TA
2 - DS was secluded within school in a separate room and taught with the TA present
3 - He did a "soft entry" into school ir through another etrance 5 minutes or so after everyone else had gone in.
4 - I took him to school and picked him up every day (he was year 5 and so had started walking to school alone)
5 - his lunch hour was managed 20 mins food, 20 minutes quiet time inside with a classmate of his choice (pretty much doing anything they wanted, computers, games etc), 20 mins in playground
6 - daily emails from TA to me copying class teacher and SEN manager giving break down of how the day went.
He absolutely hated it (and so did I and we had a week or two of horrendous backlash with teachers/TA's trailing around the school when he'd escape!
He was told very clearly when his behaviour had calmed down then he would start being reintroduced back into the wild classroom.
After about 4/5 weeks this was done gradually, a timetable was done for him at the beginning of the week showing which lessons would be done in class and which with the TA trying not to change backwards and forwards. Any backsliding of behaviour was cracked down on immediately.
By the end of about 10 weeks he was back in class (with TA in attendance and frankly really required) for all lessons except the ones he really struggled with which were the ones with outside teachers who didn't really get how to handle him at all - PE, French, music.
This was a little under a year ago - I think his total breakdown in behaviour (which had been quite rocky for a year or two before but then fell off a cliff - I think because of friendship issues he was having) started around January.
Now he is being assessed for a statement (whatever they're called these days!) so that hopefully he goes to secondary next year with a budget. He has transformed his behaviour this year in part because of the separation from the class, in part because of a concerted attempt by me to find things he loved that didn't rely on friends from school (thank you sailing!) and partly because he's looking ahead to secondary school and starting afresh.
Now at the beginning of year 6 he still has a 1:1 TA who is probably not necessary 80% of the time and the other 20% is really about helping DS deal with his emotions in the very early stages of the frustration starting so avoiding the problem. The school started taking seriously the low level "teasing" he was getting and not just assuming it was his fault. School also took on board my advice about how much positive handling he needs and even though it's counter intuitive when a child is kicking off and many many teacher do the stern telling off stuff. In fact that's the time they most need TLC. It's hard for staff and us to grasp this and dealing with it at home I know howhard it is to be positive and loving in these situations. But in the end it (often) does work but it isn;t a quick fix. Keep everyone safe, identify the triggers, remove from the triggers, let the manic brain calm down as much as possible and start building from there.
If things are really that bad, your local authority may have a class for children with severe behavioural problems which she can go to one or two (or more) days a week which can be helpful. DS was offered this but I thought the plan we had on the table should be given time to work.
He's having a great year in year 6, has come on leaps and bounds.
Sorry - that's probably a bit long but there can be light at the end of the tunnel.
Try to look after yourself - I know how draining it is.