I think it works in your favour.
If the LA cannot name any school that means your school is the ONLY one that can meet your child's needs.
The SEND Panel are not stupid.
The LA has to go with your choice of school UNLESS they have an alternative that is cheaper and which can meet ALL your childs needs as detailed in your reports/correspondence.
I presume you have had detailed reports that specify the kind of school that you have named?
I presume this includes things that NO LA mainstream or special school can provide, such as class sizes of less than 8. Weekly SALT/OT delivered flexibly across the week etc. Specialist teaching. Suitable peer group etc.
Try not to worry.
Your evidence is supposed to convince the SEND Panel. Forget about the LA. They have to prepare their case to convince the SEND Panel. If they haven't done that within the bundle they have submitted, then they only have evidence on the day.
If they have not complied with Requests for Changes and Orders made by the SEND Judge, then you can ask that that late evidence is not allowed as the LA have not produced it on time.
Do you have any expert witnesses coming with you?