All schools, all SENCOs and all LEAS are different, so you won't know what your school / SENCo can do until you've been to see her.
I am a SENCO and I do assess children, but I have done an advanced Literacy Difficulties course and am AMBDA qualified so I do know a fair bit about various assessments. But there are certain assessments that only an EP is allowed to administer because us mere mortals wouldn't be able to work out the results apparently. But most SENCOs won't have the expertise that I have, so would need to rely on an EP or possibly a SALT to do these assessments.
Until last year I used to get 3 x EP visits a year to school and had to prioritise which children were seen. It was true that any parent could request an EP appointment, but in reality these children just went onto the waiting list and if we as a school didn't consider them a priority then they never got seen.
Now I have to buy in EP visits and this year we haven't paid for the EP to come in at all. It costs us £500 for an EP visit and at the moment, because I have access to a range of assessments, I have been doing alot of what the EP used to do. Fortunately I haven't had a parent request an EP appointment this year, as I'm not actually sure what would happe then.
But in our LEA, the EPs do still have to cover Statutory work for free. I have applied for 2 Statements this year and got an EP visit each time for free as they have to contribute to the Statutory Assessment process.
It may be that an EP is not the best person to see your child anyway. It might be worth asking the SENCO if she felt that someone from the EBD team might be a better person to ask, as you have mentione behaviour traits. (But in our LEA I now have to buy in EBD services as well, so really a child now has to be quite severe before the school would justify spending that money, and I'm sure you would have heard before now if the school thought that was the case)
But, until you've had that chat with the SENCO you won't know your local (ie school / SENCO) situation is, or your LEA situation.
Mr Z - I get OT almost the next day (are they sitting twiddling their thumbs waiting for us to call I sometimes wonder), but CAMHS and SALT are very slow (currently 14 week wait for initial SALT appointment)