Also, the ->param(...) method is read-only - you can't change values of parameters with it.

Sure you can. It works pretty much just like CGI's param method. This works fine:

If you want a hash with all your params, you can get it like this:

sub some_run_mode { 
    my $self = shift;
    # note that CGI.pm does the same thing:
    my @params = $self->param; 
    my %hash = map { $_ => $self->param( $_ ) } @params;
}
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That would be a little tedious to do in every runmode, but you could do it in a prerun hook instead.