This might already be a feature suggestion. I think it is better to do the printouts in a separate tab or window since people intuitively close the window after the print is done an thus also close the website.
Topic on Extension talk:Pdf Export
There is a global variable you can set to make the PDFs pop up as a downloadable attachment instead of in the same browser window. Its:
$wgPdfExportAttach = true;
Does that work for you?
Affirmative, this works. This is a good solution. :) Thus providing a different filename on Special:PdfPrint works too now. Before it has not.