Using Schottky diodes instead of PN junction diodes for the PoE input bridges and DVDD will reduce the loss of this function by about 30%. However, there are some things to consider when using them.
The IEEE standard specifies a maximum backfeed voltage of 2.8 V. A 100-kΩ resistor is placed between the unpowered pairs and the voltage is measured across the resistor. Schottky diodes often have a higher reverse leakage current than PN diodes, making this a harder requirement to meet. Use conservative design for diode operating temperature, select lower-leakage devices where possible, and match leakage and temperatures by using packaged bridges to help with this.
Schottky diode leakage current and lower dynamic resistance can impact the detection signature. Setting reasonable expectations for the temperature range over which the detection signature is accurate is the simplest solution. Increasing RDEN slightly may also help meet the requirement.
Schottky diodes have proven less robust to the stresses of ESD transients, failing as a short or becoming leaky. Take care to provide adequate protection in line with the exposure levels. This protection may be as simple as ferrite beads and capacitors.
A general recommendation for the input rectifiers are 1 A or 2 A, 100-V rated discrete or bridge diodes.