The approach is kinda wonky but hey ... I would just pass what you need either to new and/or output. I would loosely couple the modules and rather than pass the objects, pass only the info I need:

sub start {
  my ( $self ) = @_;
  my $query = $self->query();
  my $foo   = $query->param( 'foo' );
  my $bar   = $query->param( 'bar' );
  my $baz   = $self->param( 'my_param' );

  my $start = Start->new();
  $start->output( qux => $foo, quux => $bar, corge => $baz );
}
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