Your original suggestion of edit later seems fine, but thinking about the future: What happens if they ask you to later do it live and tell you they can’t give you any more equipment?
I would just get reels of 2x Coax (BNC) + 1x audio line and run Video, Genlock, and intercom {one-way intercom would probably work, you shouldn’t need to receive signals from the cameras, simply putting a mic and small mic amp on the head end and a headphone connector [possibly with a small headphone amp, like an LM386 (this is a chip, not a premade product), and two pots] at each camera}. Run your audio seperately (two strategically placed omnis on tripods will probably get a good deal of game noise) and switch locally. If you get a digital switcher with built in synchronization, you can run just video from cameras and use 2-way radios for comms.
