I have tried this again and it didn't work —
I have connected the DIR-600 from a LAN port on the firs Linksys router, to the WAN port on the DIR-600.
The DIR-600 is running DD-WRT. I reset the firmware to defaults on DD-WRT, went in and changed the password from defaults, changed the router's LAN IP address to 192.168.1.2 (as the primary router is 192.168.1.1), and set under HotSpot Services my operator username and location ID. The only other thing I changed was the wireless settings to use auto channel and G-only. Everything else has been left as default (other than whatever enabling hotspot services changes).
After saving / rebooting, the router gets a WAN IP address of 192.168.1.109. DNS servers are inherited from primary router which is 8.8.8.8, 8.8.4.4. the SSID is changed to HotSpotSystem.com-ticowifi_3 (I even made a new location in case there was some kind of bug). I can connect to the access point via wifi, BUT it does not assign me a valid IP address. According to HSS dashboard control under locations, the location has never been seen. On the router, Power light is orange, world (internet) light is orange, wireless light is green.
Stuck 🙁