I use path_info to pass the run mode when POSTing to a CGI::App.

In mode1.html the form tag would look like

<form action='/perl/index.cgi/mode2' method='post'>
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Adjust the path to suit.

In TestSite.pm you would then have

  $self->mode_param(
    path_info => 1,
    param     => q{rm},
  );
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See the C::A docs for a discussion on path info.