An oscilloscope is the wrong tool to view ethernet data. Unless you expect signal integrity problems. Then you need the scope.
The best tool for the job is a switch with port mirror and a spare network card on a PC to run pcap with Wireshark.
You can also use a ethernet hub, but you can't buy those anymore. So unless you have from the 90's, you're better ...
No, this is not a valid test case. A valid test case would be with both jacks connected to a router or switch. You should do high data rate transmission from both ports (somehow), since when your device is transmitting over Ethernet, that is the worst case scenario.
Please also note that emissions on the Ethernet cable may arise from any circuitry on the ...
You can just chop a cable in half, carefully wire it to a breakout (keep the exposed wire straight and short as possible), and put the scope probes across either the TX or RX pair depending on which direction you want to look at.
I have done this myself to debug an autonegotiation problem, which fortunately is done with lower speed data.