So, the under-voltage issue was totally why. One of my ethernet modules has a 5V input with a 3.3 volt regulator which I had bypassed to connect to J16. I un-bypassed the regulator (so I could connect 5v), and connected 5v from J2 - problem solved.
For anyone doing this in the future, these ethernet adapters work - but don't bother trying to get 3.3 volts from the F4Discovery board - it only provides barely 3 volts. Use J2 5 volts instead if possible, otherwise use an external 3.3 volt power regulator.