Load-balanced site frequently losing permalink settings

This points to an issue with .htaccess. Going to Settings->Permalinks will recreate .htaccess, and so I think that it could be an issue with file permissions during your load or maybe because it’s a dot file the server could be treating the file differently and somehow deleting it?

This maybe useful :

I have a routine that runs flush_rewrite_rules(true) on deployment of my theme. I use wp-cli (https://wp-cli.org/) to trigger this:

// in functions.php  (using closures)
add_action('cli_init', function () {
  WP_CLI::add_command('deploy', function () {
    flush_rewrite_rules(true);
    // do something else.
  });
});

As part of my deployment process I use the following commands in my CI/CD process

cd \path\to\wp_root
php wp-cli.phar deploy

That way your .htaccess file will be recreated every time your deploy.