Perl is sending the string 02222023, but you want to send the bytes \x02\x22\x20\x23. The easiest way is to use the chr hex function:

my $send = join "", chr($month), chr($day), chr($year1), chr($year2);
$ob->write($send);
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Update: Indeed, it must be the hex function, not chr, since the numbers are BCD, not plain hexadecimal. Of course, this approach will fail in 2030, when you need to roll over from 29 to 30 instead of 2A, but that is still 8 years to go.