Finding a headless machine on a restricted network

I own a Raspberry Pi with Ubuntu on it. It's configured with a wireless card and a microUSB for the sole purpose of charge, and I need to find it on the network to connect to it over SSH.

Unfortunately, however, I'm restricted in what I can do. Hostname resolution over LAN is not permitted, and neither is ARP. All the MAC addresses are hidden by the network as well (all appearing as one). As well as this, with the start of each new connection to the network (i.e. rebooting the device) reassigns it a new local IP. There are several hundred users on said network at any given time.

How can I find the local IPv4 address of my Pi without any form of manual input to the Pi, preferably from the command line?

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