I think it’s possible to get at least some of the site onto a different SSD, however Local will still create some of the infrastructure things (NGINX/PHP/MySQL) configuration on the main hard-drive (C:\ or ~/ depending on the OS)
What operating system are you using? The permissions error you mention might have something to do with the permissions in the actual server software.
Can you take a look at the Local log and see if there are any errors there? See this Community Forum post for instructions on how to get that log: