You seem to be implying that only the last command receives the signal, but that's not true. We know the Perl script received the signal since we can observe the signal handler sending output to STDERR.

I presume what is happening is that the signal is sent to both perl and tee, and that tee exits before reading the text sent by perl in response to the signal.

If you do kill -INT 1234 where 1234 is perl's pid, tee won't receive the signal, so it will log the text perl outputs from the signal handler.