What I am curious about is whether there may be a way to programmatically pass this to the browser and have it executed the same GET statement.

That would of course depend on the browser. It's hardly a Perl question.

my $browser = "/path/to/browser";
my $URL = "http://www.foo.xx/yada/yada?query";
system $browser, $URL;
[download]

works for my browser. Other browsers may not accept remote commands. Or need specific options. Read your browsers manual page.