I agree re using some DHCP server on the LAN instead, and yes the board does have a clear hostname, but you now have to dig around in the router diagnostics to find the assigned DHCP address, which is a right PITA. There is no LCD displaying the IP.
In this case, is there some discovery mechanism, other than starting at the bottom of the subnet (what is the easy way to discover the subnet?) and polling every address up from there?
I think lots of people have been up this road before, starting from when the RJ45 was invented
and most decided to just fix it at 192.168.1.1 or some such, and then the laptop owner has the hassle of reconfiguring the laptop.
There is actually a USB-accessible filesystem with a config.ini file and in there is an option to specify the IP allocated if there is no DHCP. So maybe that is the easiest route to document.