Hi etabat,
Do you know how your server is set up to prompt for permission? If you're using .htaccess/.htpasswd there's another post in this forum where a fellow was able to set up the auth so that it could authenticate via user/pass or IP address, and he set the IP address to match the server's IP address (so it would be able to authenticate local requests, such as those made by Simply Static).
There's more info in the thread here:
https://wordpress.org/support/topic/generate-static-files-with-htpasswd-restriction
Would that work for you? If not, let me know and we can troubleshoot further.
Best,
Scott