Runner, the LA are stringing you along. The timescales aren't optional, they are set in law. If the LA EP can't assess within the timescales the LA must commission an independent one. IPSEA have a model letter you can use to complain. Let the LA know you know they must stick to the timescales even if it means an independent assessment, and if they do break the timescales you will go to Judicial Review. You should ask for DD to be assessed by SALT, OT and psychiatrist &/or CP, too.
Art and music therapy are offered by CAMHS, as are others such as therapeutic story writing and drama or play therapy. Although, as you say, you will have to fight for it. I haven't heard of animal therapy via outpatient CAMHS appts, only CAMHS inpatient/day units or medial wards, however I don't really know much as DS1&3 are allergic. All of them can be included in EHCPs which can guarantee them (if they are in section F as they should be). If CAMHS won't or can't provide the LA must commission independent providers.
Hyper focusing on one negative event is common in those with MH conditions. I often wonder what comes first. DS1 holds a grudge for a very long time, you don't get a second chance with him. I remember Zoo posting her DS is similar.
I would be wary of moving schools at this point without an EHCP. You could be jumping out of the frying pan and into the fire. Staff that don't know DD, in a school DD doesn't know, without support in place from the beginning could set DD up to fail.
It was being in crisis on the brink of hospitalisation that got DS1 the help he has too.