Then the supply before the diode supplies 3.3V + the voltage drop; for example 3.6V.
So now you need a 3.6V power regulator, in additition to the 3.3V one you likely already have for all the components built for 3.3V.
Or another diode to put in-line with the DP circuitry. It's probably not ideal from a low-power perspective but if that's what you gotta do to make robust and compliant hardware, you do it.
A Schottky will drop .3V; That's within the range (±10%) specified in the specs
Yes. But now you have given up all of your tolerance so you better have a perfect (or over-voltage) power supply. This means that you now have less choice in power supply.
As a parent commenter said, you could also put a diode for your circuitry. You would then have a single 3.7V supply.
And 10% higher power draw just for this one feature. No, there are better ways to solve this.