Probing ethernet cables with a scope
No, you can not assume there will be no signal on the line just because you're not deliberately sending something at the network packet layer.  There are things called link pulses, and there can be other negotiation going on between the switch and the PHY and possibly the MAC.

The best thing to do would be to let the MAC and PHY do their jobs.  You can use a microcontroller with a MAC and PHY to blip a pin whenever one of your packets is received.  You'd write a very bare "driver" that just waits for a packet at the MAC layer, blips a pin, then clears the packet.  With such dedicated firmware, the jitter it adds should be quite small.

If you are willing to slow down your packet rate, you can do this even more easily.  Most switches and other ethernet devices have send and/or receive LEDs.  These are often blipped for a few 10s of ms when a packet is sent or received, which is why sending one every 5 ms won't work.  If you send a packet every 100 ms, the LED should be off again when the next packet arrives.  You can also wire up your own PHY and set it up to blip a pin a short time for each packet.  That's basically what the chip in the switch is doing when it drives the LED.