100Base-TX uses scrambling on the wires, so the line is always humming with activity even during idle periods.
Personally, I'd get an Ethernet hub, or a switch with port mirroring capability. (Many managed switches can do this; you just need to dig around in the UI to find that feature).
Then, plug it in between the two devices you want to monitor, and attach the mirrored port to a separate Ethernet device that you can probe.
On that device, look at the MII RX_DV pin with the scope.
You still don't know the latency through the switch, but it may be small compared to the times you're looking to measure.
I'm not sure whether it's possible to passively sniff a 100BaseTx line. Can't hurt to try, I suppose.