I was able to confirm today that Last Comments 1.3 does indeed work for invited readers of private blogs!
Your invited readers will need to be logged in to their Blogger dashboard before it will work. If they are logged in to their dashboard, they will see their email address and at least the "Dashboard" and "Sign Out" links in the Navbar in the upper right corner of the screen while reading your private blog, and if they get that far, they will be able to go through the authentication process for Last Comments 1.3 just like a blog owner would, and then they will be able to load and view the latest comments left on your private blog.
Users of Firefox might need to enable third-party cookies and all users should be aware that Last Comments 1.3 makes use of a pop-up window for the authentication process, which might get blocked by pop-up blockers on your browser.
Update: Internet Explorer users may have to click on a blog post or refresh the blog after the login process to retrieve the blog's comments. I am currently looking into this strange behavior.
P.S. to jakub kovařík, I left a comment for you about a widget I put together to get the most recent comments from the feeds you listed for your website. Email me or continue using the comments here if you have any questions.