is it working with node access module?

This performs a drupal_http_request which basically remotely renders a drupal page. This means that it's rendering the content as an anonymous user so it'll be grabbing and displaying whatever that ends up being for the anonymous user. This project has been used with the most success in an offline, local-host setting. It your looking at this for speed reasons then you should probably check out http://drupal.org/project/boost . This project was designed for having offline copies of a website or in instances where a shared hosting environment wouldn't let you have Drupal installed but you still wanted to use it / manage it.

Being able to run HTML Export with a defined user would be a great help for capturing different views of a protected site. This is not the same as making everything public (on a local copy).

it's in my queue and I realized a way to do it but it's pretty low on my priority list at the moment. Still, it is in the roadmap for a system I'm working on so this will be added, just don't know when.

So, what would be the way to do it?

download ELMS and reverse engineer the way of accomplishing this. There is a export capability derived from HTML export that allows for group based publishing to HTML. It'll be some time before it's created as a module on its own if it is. The method used in the version in ELMS uses the native drupal page render function so it's just based on whoever is logged in as to what they see. Workflow I'd see is masquerade as another user (in code) w/ that role, render the pages they have access to on the site, save to .tar and then unmasqurade. https://drupal.psu.edu/content/216

In what way is this fixed?

functionality is now in place to masquerade as another user which effectively mirrors their node access rights per a certain role. It is marked experimental functionality but is included. You need to install the masquerade module for this to show up as an option.

Great news, thanks!