Personally, I like to use Carp::croak over die because it reports the problem on a line number that the module user can affect, instead of reporting the problem within a module that the module user will not edit/change.

I think your approach of having a reporting scheme that stores the error message and returns undef or (if "autodie" is set) croaks is the best approach:

sub _error {
    my( $self, $message ) = @_;
    $self->{last_error} = $message;
    croak $message if $self->{autodie};
};

sub do_something {
    my $res = eval {
        $webhook->execute(...);
    };
    if( my $err = $@ ) {
        $self->_error( $err );
    };
}
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