Yeah, what a badly written piece of HTML, groups of option buttons of which only one should be selected should have the same name.

The WWW::Mechanize documentation is unclear, but have you tried button1 => undef?

$mech->fields(button3=>'button3', button2 => undef, button1 => undef);
[download]

or

$mech->submit_form(
    fields      => {
        button1 => undef,
        button2 => undef,
        button3 => 'button3',
        }
    );
[download]

should work. Also, you don't need to quote hash keys that are alphanumeric when you use the fat comma (=>).