The thing to keep in mind here is that an iframe runs on the client side. It is basically a browser within a browser. This also means that it will use an existing session if the client is logged in on the site (you already discovered this):
When logged in with my regular user on that other site, the iframe was
displayed for my regular user.
The problem is that Drupal runs server side. Theoretically you should be able to use
drupal_http_request() to send the login data to the other site, catch the session cookie and initiate a session. But that would initiate a session on the server, not on the client. The same with cURL, it would log the server in, not the client. So the iframe for the client would still show the login button.
Can I trigger the pressing of the login button and then fill in the
login form, or is it possible that that is not possible?
So I will conclude with no. There is unfortunately no sensible way of doing this.
Finally there are 2 more things to keep into account;
What if the client happens to already have an account on the site? You would override his session and he would be you. I would personally hate your site if it forced me to logout and in again on another website.