How do airlines handle uncertainty in passenger weight for weight and balance calculations?
They don't handle that uncertainty.

Safety is always the prominent factor in the aviation industry. That said, measurements need only to be "accurate enough" for the purpose of safety. Being off by small amount would not compromise safety.

Furthermore, the atmospheric condition is always different; it is so dynamic and cannot be precisely measured. As long as the measurement is accurate enough, the aircraft will fly.

Theoretically, you could argue that there's a chance:


calculated weight is already at MTOW
every passenger happens to weight way above average
they all bought a lot of souvenir at the airport shops
density altitude was extraordinary high
a 10 knots tailwind was present at takeoff


and the aircraft overrun the runway during its takeoff roll. However, such a chance is so small that we accept our current method as it is and expect the aircraft to tolerate that uncertainly within its designed safety factors.

Such believe was found to be wrong when a small aircraft crashed on takeoff in 2003. Regulations were then modified so that on a small enough aircraft, the actual weight of each passenger (plus luggage) must be used, rather than using a standard weight for estimation.