No, that's not accurate.
From what I have seen in documentation on DHCP it is a first server to answer wins sort of protocol. So you can have 2 DHCP servers on the same subnet if they hand out IP addresses from different ranges.
For the OP's configuration I would do similar to the following:
Rack 1:
Router IP 192.168.1.1
WLS units get static IP 192.168.1.11 to 19
Hand out DHCP 192.168.1.110 to 119
Rack 2:
Router IP 192.168.1.2
WLS units get static IP 192.168.1.21 to 29
Hand out DHCP 192.168.1.120 to 129
Rack 3:
Router IP 192.168.1.3
WLS units get static IP 192.168.1.31 to 39
Hand out DHCP 192.168.1.130 to 139
Also have a label listing addresses 192.168.1.200 to 250 as safe static IP addresses.
The router listed has auto uplink ports so just one cat5 rack to rack and plug the laptop into the most convenient rack. Do not use the WAN port on any of the routers.
Having said all that, I think that having a "no network config necessary" is nice but not totally needed. If a tech can't set an IP address on his or her laptop do you really want them to have access to your wls system?