TP-link power splitter I managed to find seems to be non-isolated. Bad thing for you is that PoE usually have diode bridges on input (to be polarity-independent and work on straight or X-cables) meaning that if your 48VDC supply is not galvanically isolated and for instance it's minus terminal is grounded, sparks can fly if you miss to crimp ethernet cable right way. Try to swap 4,5 and 7,8 pairs, measure ground potential difference again.
Of course, for careless PoE usage best solution is to use isolated splitter.
The problem you are experiencing is due to the resistance of your umbilical. You have a transmission line. The \$I^2 R_{Cable}\$ losses are eating up your power.
Measure the voltage and current at the terminals of your motor. As I increases \$V_{Load}\$ will decrease.
There might be enough voltage to start at no-load (especially if rov is out of water - pushing air) but as current increases, voltage drop to wire increases, which explains your observed characteristics.
Solution would be to send ac at a higher voltage to rov, step it down in the rov and drive motors.
Edit...
Or increase the gage of your power feeder cable as suggested by: Tether - Power and Voltage Drop Examples.
It really depends on your current. 6 motors can draw a lot especially going from locked rotor to full speed. 20A is a lot depending on length of your umbilical.
Look up resistance of wire from AWG (American Wire Gage). They have R/100ft. Multiply it by your tether length * 2 / 100ft.
The 2 is because you have to get current to load and back. Think about number of pins on a lamp.
Or you can also calculate resistance. $$R_{Cable} = \frac{ρ l}{A}$$
ρ = 10.37 Ω CM/ft, l = 2 * length of your umbilical, A in CM from AWG.
$$V_{DROP} = I R_{Cable}$$
Components of an ROV system
Best Answer
Sorry but your circuit won't work. As Michael says Ethernet is too sensitive for solutions like that. So here are a few other suggestions that might help.
The simplest solution would be a managed switch that both machines were connected by and then use the Pi to connect to the management interface, via serial or telnet, and tell the switch to close and open the ports as you need.
Or write and install a service on the hosts that when the authenticated Pi sent a message to them the services used operating system functions to disable the network interfaces. Obviously there is a problem with this solution when it comes to turning the connection back on.
You might be able to use a USB Ethernet adapter on the Pi to create your own two port switch, then bridge the connections using the Pi's OS. The Pi can then enable and disable the connections at will, but I'm not sure what the performance would be like.