You could request an EP assessment. If you are going to do that, you should do it now. However, the LA may say no and force you to request a reassessment of needs which they could also refuse but would have the right of appeal. A reassessment of needs wouldn’t be the right route for you. It may not change anything and still leave you needing to appeal but having wasted months.
An LA EP report is also unlikely to be sufficient for an appeal for a residential SS.
In your situation, an EP would be better than a CP. If you do go ahead with the CP, make sure they have SENDIST experience. The majority of CPs do not.
You could try: Jemma Levy, Ruth Birnbaum, Vivienne Clifford, Mike Hymans, Patsy Kershaw, Marina Gouzouli-Allen, Siobhan Aubin, Paul Kelly, Stephanie Warman, Jonathan Middleton, Leanne Short, John Pugh.
There is a wait for all, but they are good and you will have a wait for SENDIST anyway.
Normal physio and OT referrals are unlikely to lead to sufficiently detailed, specified and quantified reports. OT in particular can help for a residential SS appeal. Not every provision could deliver the level and type of OT input (direct and indirect) and environment many requiring a residential placement need.
Obviously, I haven’t seen the evidence from your social worker, but unless the evidence from your LA is unusually good, an independent social worker report will be superior.