Based on the feedback here (and thank you!), I now understand that "Pipe to Program" cPanel email filters are unable to pass their input on to further filters. The "Pipe to Program" filter eats up the message, and no other subsequent email filtering or processing takes place.

Re-emailing the incoming message from the "Pipe to Program" filter is not an option for my particular use case.

So, I see now that I'm totally out of luck with this approach.