The problem was solved without clicking the button!

In perl I had to open https://example.com/site/content.htm
Then I got a page with a button that pointed with a href to that same link.
I had already tried to open that page again without the button, but I got the page with the button back again. Caught in a loop.

The trick I applied was getting an upward link: https://example.com/site/

This gave me the same page with the button and te same href.
But now opening that href worked!
No need for mechanize to click the button anymore.
But I will try the solution given to that as well.