I've never tested this, but what about foregoing the haunt roll and instead triggering on the nth omen? You'd still be at the mercy of the room tile stack, though. If you really want to control when the nth omen is found, you could seed the room tile stack so the omen rooms are equally distributed.
Alternately you could stipulate that after a certain amount of time, the next omen automatically triggers the haunt. Still make the haunt rolls before that point, but if you've been playing for 10-15 minutes and are still exploring, make the next omen automatically trigger. It should give you sufficient time to play through the haunt portion.
Great ideas, we might try one of these options tomorrow. If we do I'll try to remember to report on how it goes. Thanks!