Hi Ian,
I'm not sure if you're using .htaccess file or not ( I tried to login with provided FTP account but it did not work), if you have this file, pls backup and go to Backend > Admin Tools component > .htaccess maker and add 'json' file for exception > Save and create htaccess file.
It seems that Admin Tools is blocking json files so that the compilation doesn't work. If you use Inspect element editor when saving template style, you can see the 404 & 500 errors-- which comes from server side or 3rd party extension on your site.
Or another way is disabling Admin Tools component temporarily to check first.
Regards