With that said, whom should I open a bug report with?

Net::Ping is a part of perl core, so you can send your bug report as an email to perlbug@perl.org .
AFTERTHOUGHT: I just noticed that perl core currently (perl-5.27.7) has Net-Ping-2.62 - so you probably should check and see what its behaviour is prior to reporting.
UPDATE: I finally got around to checking the behaviour with Net-Ping-2.62, and it does have the same buggy behaviour as you reported for 2.63.
Most people would file that email by running the perlbug command - as that ensures that all relevant information about your system and perl itself are included in your report

Note that there are 2 ways that this bug could be fixed:
1) Restore the previous behaviour of ping();
2) Alter the documentation of ping() to match the current behaviour.
So it's not guaranteed that the behaviour of ping() will be reverted to its earlier form.
However, back-compatibility considerations should provide an incentive to go with 1)

Cheers,
Rob