just talking out loud here... there are no breakers tripping so there's no overload on the breaker, and there's no gfci tripping so there's no ground fault either... but your problem could be related to some overload since you said you've got to have all the lights/spa pump/heater going for it to happen.
this is just wild conjecture but it's possible the board is experiencing a "brownout", which occurs when the supply voltage to the micro-controller dips low enough (below 5vdc) that it actually shuts off for a second and restarts. It could occur for a variety of reasons... could be the ac voltage to the system transformer itself dips, or the load on the transformer is abnormally high - possibly from bad relay drivers - pulling down it's output). But if it is a brownout that's happening, when the micro shuts down the relays would also shut off - which would in turn clear the load on the system and allow it to come back online.
unfortunately these things can be a pain to diagnose... I'd start by watching the outdoor panel for any indication that the board is getting reset when the fault occurs... see if the display cycles off/on (or the backlight) when the fault happens. Or, before the fault occurs, get into some menu setting... if you get kicked back to the main screen you'll know the board was reset.
If those tests don't tell you anything you'll probably want to measure the current draw of the electronics (10vac supply) and relays (18vac supply) to see if there's anything abnormal. Then try to reproduce the fault and monitor the supply voltages to the board when the fault occurs. I'd probably also monitor the output voltage of the onboard 5v regulator to see if it drops below 5 - but to do this kind of monitoring you'd need a multimeter that can either do min/max or data logging.