It hands out the private IP addresses automatically to every MAC address that is broadcast on the network
Actually, that's not how a DHCP server is supposed to work. If it is doing that then the server is malfunctioning and should be replaced. It is entirely, 100% valid to have devices on the network that are not DHCP-based. It is wrong for the DHCP server to allocate an address to any device just because it's on the network. It should only assign addresses to devices that ask for one (via a DHCP request).
As such there is nothing (or, at least should not be anything) that prevents you from assigning a spare IP address in your private subnet to the LOM interface. The only question is how to work out what address to use. For that you need to know the DHCP range that the server is using.
For example, if your private network is 192.168.1.0/255.255.255.0 and the DHCP server is handing out addresses from 192.168.1.1 through 192.168.1.100 then that leaves all the addresses from 192.168.1.101 through 192.168.1.254 (less the address of the router itself) as being free for statically-assigned devices in your network.