If you try to update the T3 plugin from the Update Manager, you will always get this error:
Warning: Update path does not exist. Message: Error updating COM_INSTALLER_TYPE_TYPE_
This is caused by a bug in the T3 plugin that has not been fixed by the T3 developers.
Unfortunately this is out of our control. You can only update the T3 plugin manually by downloading it from the Github repository at https://github.com/t3framework/t3/tags.