So did you appeal? If yes, when is your hearing and have you appealed BFI? If no, do you currently have the right of appeal?
If there is no placement and no type of placement named in section I of the EHCP and you are not EHE, have you asked the LA how they intend to provide the provision in F and a full-time education? If they intend to make arrangements via a school, they need to follow the correct process and amend the EHCP to give you the right of appeal.
I would check the LA has not recorded DS as being EHE.
Unless there is more to your circumstances than your posts alludes to, the LA cannot force you to send DS to a school as things stand at the moment. Even if the LA name a school in section I, this doesn’t automatically register DS at the school. Although if you don’t intend to EHE or make other alternative arrangements yourself, you should take steps to rectify the situation, e.g. via appealing and seeking alternative provision from the LA. However, the LA could go down the school attendance order (SAO)/Education Supervision Order (ESO) route if they were not satisfied DS was receiving an education at school or otherwise.
The fully independent school cannot be forced to admit. You would need an offer of a place.
Have you looked at all independent schools within travelling distance, including those out of area? The maximum recommended travel time for primary is typically considered 45 mins but some travel further.
Most ARPs are not separate registered institutions. Therefore, the mainstream school is named in section I and the provision provided by the unit included in F. You need to check if the ARPs are separate registered institutions or not. The schools with ARPs can be named even if the school objects. On its own, being full is not enough of a reason to refuse to name them if they are your preference.