You can skip the Hap lite from the chain easily. (once Edgerouter runs the hotspot)
Once the Edgerouter is giving the splash page, all APs should be used in transparent bridge/AP mode.
Linksys E1200 can be used by a disabled DHCP server (even factory firmware or DDWRT 21061 for stability) and connected by its LAN ports (when by default bridged with its wifi) to the Edgerouter via switches.
Steve Hars