List of Web Hosts that require 644 .htaccess file permissions – 404 permissions are not allowed on these Hosts and will cause a 403 Error
If you Web Host does not allow locking of your Root .htaccess file/404 file permissions for your Root .htacces file, then go to the BPS Edit/Upload/Download tab page and click on the Turn Off AutoLock button. This will prevent your Root .htaccess file from being automatically locked when you upgrade BPS, which will prevent a 403 Error from occurring on your website.
If your Host is LiquidWeb.com then this known issue exists above for your host. There are several hosts that use "liquid" in there hosting name so your host could be a different host.
There are a few possibilities here since you are getting the error message that you do not have sufficient permissions.
1. Your folder permissions are set too restrictive. Make sure that your folder permissions are set to 755 and file permissions should be set to 644. Very likely.
2. You have set group ownership permissions that are blocking you from being able to write to folders. Not likely, but possible.
3. Your host is using open_basedir and a restriction is blocking the creation of the root .htaccess file. Not likely, but possible.
4. You have the 404 permission problem on your host that I quoted above. Not likely, but possible.
5. Another plugin is causing this problem. Deactivate all other plugins and see what happens then. Moderately likely.