If you sign in on a server (including A365) on Altium Designer, you won't see the three dots. You need to sign out from server for the three dots option to become available because it's for local folder. Please kindly click on the cloud sign next to your name on the top right corner to sign out from A365 first and try again.
The behaviour is documented here: https://www.altium.com/documentation/altium-designer/managed-bom-templates-ad#!reusing-a-managed-bom-template-item
If you feel, that should change and that the option to choose local templates should always be there, also if logged into a managed server, please vote on this bugcrunch entry: https://bugcrunch.live.altium.com/#Bug/13822