Hello KENNINGHAM-BROWN Adam,
Greetings. Thank you for reaching out.
I understand that you are having trouble restoring a deleted channel in teams. Based on your description, you have checked the permissions for the application, please confirm if members are allowed to restore in the teams settings like in screenshot below. If so, please check whether any of the other members can restore this channel.

->Also, how long ago was this channel deleted? Please note that channels can be restored within 21 days of deletion (soft delete). If you deleted these more than 21 days ago, these can't be restored as they are now in a hard delete status.
If these channels were deleted less than 21 days ago and none of the other team members can restore the channel, you will have to reach out to our online tech support team to check further.
As an organization admin, you can raise a service request through the O365 admin center if you have the permissions. Otherwise, you can reach out to your M365 admin to raise this for you. A support agent can further connect with you remotely, for further troubleshooting and involve the backend team which might be needed to restore the channel.
Get support for Microsoft 365 for business -- Online support.
Your patience and cooperation are much appreciated.
Best Regards,
Ashraf.