is it possible to ask a browser to print using the screen CSS rather than the print CSS?
I don't think any browser offers this option as default.
Using a developer tool such as Firebug (for Firefox) then you can edit the live HTML to change the appropriate
media attributes on the
<link rel="stylesheet" elements. You can then print using the screen CSS (or whatever CSS you like).
However, if the developer has specified a print stylesheet then it's usually to make things better. If you were to print using the screen stylesheet then it might not appear like the screen version since it might be reliant on background-images (which don't print by default), amongst other things. So, if you want to print exactly what you see on the screen, you're probably better off taking a screenshot.