This is bad behaviour by the LA
For a phase transfer they should name both schools see Code of Practice 8.132 below.
If you give me a few minutes I will see if I can find the date for naming both schools for transition from pre-school to primary
Currently a good special school wants her - if this is a maintained school whether it is maintained by your LA or another LA then you should get it unless it does not meet her needs or is incompatible with the efficient education of others or is an inefficient use of resources.
They do have to consult the governing body of the school and also the neighbouring LA if it is not one of their own. Have a look through Chapter 8 of the code. 8.63 comes to mind, That would be a reasonable excuse for delay, but the rubbish they are coming up with is just an outrage.
BUT as it is clearly an appeal that is a transition appeal, I would just file a request for changes at Tribunal asking for an order that the LA name their chosen school. With indie reports in the offing you will want the experts to visit both schools (if the LA does not agree with your choice) and so you should give those reasons as a reason for needing the order.
You could attach the letter from the LA as long as it is not from mediation which should be kept confidential.
By the way, if the letter were a genuine attempt to settle it would be privileged and you could not disclose it to the Tribunal - have a look and see if it says 'without prejudice' at the top. If it doesn't, I think you can disclose it to the Tribunal. If the LA protest, say that it obviously wasn't a genuine attempt to settle. I think the Tribunal will be outraged by their disregard for the rules.
Good Luck
Oh and if you post under the child tab and in the 'chat about you child' thread you will get far more replies - all the action is over there.
8:132 When a child is moving to a new school, particularly at phase transfer, the statement should be amended to name in Part 4 both the current placement and the new placement, stating an appropriate start date for the latter. This will make sure that parents, children and the receiving school can plan well in advance of transfer, and entitle parents to appeal to the SEN Tribunal in good time if they disagree with the named school. At phase transfers, except from early education settings to the primary phase, the statement must be amended no later than 15th February in the year of transfer. The amended statement should specify the current school attended by the child (if any) and the new school, indicating the date on which the child will move there.