On the DotVoid.com blog Danne shares a quick tip on forcing downloads over HTTPS to Internet Explorer (which, of course, has to be difficult about it).

The problem is that Internet Explorer does not handle file dowloads without caching over https very well. Or at all. According to knowledge articles on Microsofts website the problem occurs when having one or two of the http headers. [...] Previously I have have just omitted the http header "Pragma: nocache" for IE but it seems it does not always help.

So the fix is relatively simple - rely on the $_SERVER['HTTP_USER_AGENT'] value to tell if it's an IR browser or not. If it is, "Pragma: cache" works. If not, stick with "Pragma: no-cache".