Ok. Just realised that there is 1 more major issue.
My blog is installed on mysite.com/blog while the with your suggestion of keeping the same WP for both static site + blog, I would have to shift it to mysite.com This way I will loose all my existing links, which are many.
So, is there any way I can keep :
Statis ages at mysite.com/%pagename%
Blog posts at mysite.com/blog/%postname%