Use tcpdump or wireshark to look for an ARP packet when the machine starts up. I don't know if yours will spit one out, but it might.
Even if it does reveal its MAC, the machine may not support DHCP. Convince your network admin that you need a static IP, because it sounds like you do. DHCP can coexist without conflicts, and MAC as a form of access control is very weak (can easily clone them), so there is no technical reason why you can't have a static IP without a MAC restriction (the MAC restriction can also be applied after the machine is online).