I would think that I should be able to use the IP address in the URL (http://1.2.3.4/) and force a Host header in a LWP request somehow, but after looking at the LWP documentation and trying several different approach, I've not come up with anything that works.

That was my first thought as well, and it seems to work just fine for me:

$ua = new LWP::UserAgent;
print $ua->get("http://1.2.3.4/", Host => "www.domain.com")->content;
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