Do you have two different power sources feeding the two breakers or just the one that is coming from the 60 amp in main panel?
If both the 60 amp breaker and the 30 amp breaker are fed from one 60 in the main panel, the one in main panel will trip when that third element which is fed by the 30 amp breaker comes on.
If your furnace is being fed by one 60 amp breaker, when the third element comes on, it is drawing more than 60 amps. In the unit, everything is protected because only 2 elements are on the 60, but when the other element comes on, it is drawing on the same breaker in main panel. You will need a separate 30 amp breaker in the main panel supplying voltage to the 30 amp in air handler or a larger wire and breaker to feed both the 30 and 60.
Does this make sense to you? You have a 60 amp circuit from your main breaker panel feeding a 60 amp breaker in the air handler. This is fine and will not cause any problems. However, when you connect the third heat strip ,on the additional 30 amp breaker with the 60 amp circuit, you are overloading the circuit and therefore tripping the breaker. To prevent the circuit from overloading, you will need to add a separate 30 amp circuit to the air handler to feed the 30 amp breaker and third element. Or you would need to replace the 60 amp breaker to a 90 amp breaker and replace the 60 amp wire with 90 amp wire.
Let me know if you have a separate 30 amp breaker in the main panel that supplies power to the 30 amp breaker in the air handler. If not, you can turn the 30 amp breaker off and just use 10 KW heat until you can have wiring corrected.