In the commandline app the line
print $bitly->shorten(URL => shift @ARGV), $/ while @ARGV;
should be changed into
print $bitly->shorten(URL => shift @ARGV)->{url}, $/ while @ARGV;
or even better
for ( @ARGV ) {
my $url = $bitly->shorten( URL => $_ );
$url = $url->{url} if 'HASH' eq ref $url;
print $url, $/;
}