After searching some more, and taking a hint from Oleg's answer, I've found the method HttpMethodParams::setVirtualHost().
when HttpClient formats a request, it always creates the "Host" header itself just before sending the request - so it cannot be overridden as a standard header. But before the host name for the "Host" header is generated from the URL, HttpClient checks the HttpMethodParams object to see if the user wants to override the host name. This only overrides the host name and not the port so it would be easier to use, though not as intuitive as I'd like.
The code to use this can look like this:
Method m = new GetMethod("http://some-site/some/path");
m.getParams().setVirtualHost("some-other-site");
client.executeMethod(m);
Because I like one liners, this can also be written as:
client.executeMethod(new GetMethod("http://some-site/some/path") {{
getParams().setVirtualHost("some-other-site"); }});
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