Have you considered eliminating calls to header.php, sidebar.php, and footer.php from wordpress’ index.php, and then using iframe on your html page to include the index.php page on your site?
Another approach could be to add the html code to index.php directly.
May not be the best approach, but it could work, depending on your situation.
It’s a matter of opinion, and I’m not sure what your current template really does, so I could be off base, but I would use WP in it’s entirety, but only for the blog section. I would find a theme that is sort of close to my template, then create a child theme and style it so it exactly matches the template.
You can design your site to not use themes, by setting the WP_USE_THEMES constant to “false”. There is some more information in the Codex: http://codex.wordpress.org/The_Loop
Here’s another link with even more information that will probably be relevant to you: http://codex.wordpress.org/Integrating_WordPress_with_Your_Website
