Assuming the independent school isn't named in the statement/EHCP, the LA wouldn't have automatic access. However, it would make sense to file the reports with the tribunal, as they will be highly relevant.
If you got the reports with a view to litigation ie tribunal then they are privileged which means they are confidential as between you and the person you asked to do the reports. This is litigation privilege, If the LA demand to see something privileged then you do not have to give it up.
Privilege works document by document, so if the report that you don't like is a paragraph or two in a larger report, you can either claim privilege for the whole document, and release none of it, or waive privilege and release the whole thing, warts and all
Independent experts often say that a report has been requested for tribunal, which is then 100% clear that it is privileged/confidential until the parents choose to release it ("waive privilege"). The document is the 'open'.
It is less clear where a school has been approached for a report before an appeal has been filed, but if the context of your request to the school was that you would be placing her there via a statement of SEN, then I think it is worth arguing that it is privileged. It might be worth making this clear to the school in writing that you take the view that it is privileged, and that you will let them know when, and if, you waive privilege.
Privilege also attaches to any attempts to settle the case before Tribunal, so that you can make offers and have meetings without the other side having to worry about the Tribunal panel finding out.
The other type of privilege is between solicitors and clients where all correspondence is privileged and cannot be demanded by a court/tribunal. ditto barristers and clients.
Sorry this is a bit of a lesson. I have just read it back and fear I should have started 'Good Morning Class. Today we are learning about Privilege'