Naming a CIC in Section I

My daughter has severe SEMH needs and we’ve identified a CIC which delivers alternative provision for children with SEMH, however they only offer part time hours 15 hours over 2.5 days. The LA are saying as the organisation is not a school, they cannot name it. The organisation have said that they have other kids who attend with EHCPs and this is funded through a personal budget. The other 10 hours to make up full time provision we are considering another provider which isn’t a school but an alternative provision provider. We’re anxious to know whether we can have named in section I of my daughter’s EHCP the CIC and also if we can also have named the other provider for the other 10 hours?