Not that I don't admire the evilness of the technique, but wouldn't it be easier to do something like this?

{
    my $old = \&HTML::TableExtract::TableState::_taste_text;
    no warnings;
    *HTML::TableExtract::TableState::_taste_text = sub {
        local *HTML::TableExtract::TableState::_taste_text = $old;
        eval { shift->_taste_text(@_) }; 
        print STDERR $@ if $@; die $@;
    };
};
[download]

This would only fall down if _taste_text depended on the state of the call stack (which you could get around with Sub::Uplevel), or does different things if not called as a method (which you could get around with called_as_method from Devel::Caller).