General guide for anyone else having the issue:
Before you go on fixing permissions we will need to make sure any plugin isn’t interfering with the live customizer. It doesn’t matter if you have one plugin or ten, steps are the same.
First disable all plugins and then check the theme options if they are saving, if they are saving then it means some plugin was causing the issue, to find it simply activate plugins one by one and checking theme options.
You can enable all your plugins once you are done with the theme options.
After verifying the above and ruling out plugin issues, ask your host to reset file system permissions.