Depending on the click tracks, you may be able to use the custom click track from each song as a timecoded track to sync to.
If midi is your only option, most lighting desks support show control via Midi. Failing having a desk, You can get Midi inputs for various lighting software fairly cheaply.
If you have the budget and time, I'd be recording their click tracks as mono click and timecode. Let them trigger it, but feed the timecode to where you can insert it into your desk. From there its just a case of programming the show as fully timecoded.
If I remember correctly, the Road Hog 3 doesnt have any manner of timecode input on the desk, or MIDI inputs.
That said, there is talk around on forums of using MIDI to usb converters, these may work for you, you would then need some sort of MSC (MIDI Show Control) source to trigger the Hog.