This isn’t a full answer but might get you pretty close to real/robust date handling with just a little bit of trivial Hebrew formatting, via hashes, left to do. I recommend suggesting a patch or bringing up the issue of “native” formatting to the module authors; DateTime::Calendar::Hebrew.

#!/usr/bin/env perl
use strictures;
use utf8;
use DateTime::Calendar::Hebrew;

my $count;
my %months = map {; $_ => ++$count }
    qw/ Nissan Iyar Sivan Tamuz Av Elul Tishrei
        Cheshvan Kislev Tevet Shevat AdarI AdarII /;

for ( <DATA> )
{
    my ( $day, $month_name, $year ) = split;
    my $dt = DateTime::Calendar::Hebrew
        ->new( year  => $year,
               month => $months{$month_name},
               day   => $day );

    print $dt->ymd, $/;
}

__DATA__
15 Elul 5776
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5776-06-15
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