Why is this Schotty diode breaking a PoE PD module board?
That's a bad idea - look at at the schematic on page 3 of your datasheet: Placing your diode where you do disable C35's ability to flatten the ripple of the half-wave rectified current induced in the secondary side of the transformer.
The primary side senses that the energy in the transformer core isn't drained, and consequently shuts down. Good device!
Also, when looking at that schematic, you'll see that your reverse voltage protection does nothing useful, as there's already a diode in place, directly at the transformer.