There are a few different issues to unpick.
When the EHCP is finalised, you will have the right of appeal. You can appeal to SENDIST if you wish.
You mention 1:1 is being provided now, but you don’t actually mention 1:1 is in the EHCP. Is 1:1 detailed, specified and quantified in F of the EHCP? If not, regardless of the school named and who is responsible for the fees/SEP, the LA does not have to ensure 1:1 is provided.
Similarly, does section F include small (defined) class sizes?
If the school is named in the EHCP, the LA is responsible for the fees as per section 63 of the Children and Families Act 2014. To get an independent school named in the EHCP and fully funded, you will need to prove the LA’s proposed school(s) can’t meet needs &/or it isn’t unreasonable public expenditure. Conversations about the cost of a place are common. I suspect this is why is the LA will agree to name the school if there is an agreement that the independent school will meet costs.
Sometimes LAs will name a school in section I and write something about it being parental preference, parents funding the fees and they believe needs can be met at X school instead. If that happens, you need to appeal when finalised.
When the school isn’t named in section I, some LAs will come to an arrangement whereby the parents pay the fees and the LA funds the SEP, the LA doesn’t have to do this. They often say parents are making suitable alternative arrangements thereby relieving them of their duty.
Non-wholly independent schools can still be named even if they object when consulted and can be directed to admit.