Hi @David Cook - Adaptable IT ,
Welcome to Q&A forum!
The "Move to" and "Copy to" options in SharePoint Online are indeed not related to downloading files locally, so they should theoretically be available even with limited web-only access.
Based on your description, this should not be caused by insufficient permissions. Therefore, it is worth double-checking whether there are any specific settings in Conditional Access Policies that may have different effects on unmanaged devices. If you are not sure, you can share it here and we can discuss it together.
Alternatively, you can use Power Automate or PowerShell as a workaround to copy and move first.
https://blogs.iuvotech.com/how-to-move-files-in-microsoft-sharepoint-using-power-automate
https://www.sharepointdiary.com/2018/03/sharepoint-online-move-files-using-powershell.html
Non-official, just for reference.
Good day!
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