How airworthy was the BV 141?
Why would you think the weight was not correctly distributed? If the center of pressure and the center of gravity coincide, there is no problem. Every pilot needs to bring the center of pressure right above or below the center of gravity when flying; this is commonly known as trimming the aircraft. Almost all aircraft have small asymmetries which need to be counteracted by small control surface deflections or trim tab settings.

Note that the fuselage of the BV-141 is offset to the left and the cockpit is offset to the right. This keeps the spanwise center of gravity location near the center of the wing, so any resulting rolling moment is small. What remains (maybe due to an unusual high or low loading of the cockpit) can be trimmed away with the ailerons. A yawing moment due to asymmetric thrust can also be corrected by rudder trim, and the engine was placed on the left so the p-factor of the propeller (which creates more yawing moment at low speed) would counteract the yawing moment due to its out-of-center position.

The BV-141 was designed as an observer platform with an unrivaled field of vision. By using an offset fuselage, only the left side was obstructed by an engine or tailboom. If you compare the BV-141 with its competitor for the same tender, the Focke-Wulf 189, the advantage should become obvious.



Focke-Wulf Fw-189 (picture source)

Note that the BV-141 was a private entry in this contest and was not selected.

A similar aircraft with an asymmetrical mass distribution was the Flyer I of the Wright Brothers. Here, the right wing was 4 inches wider than the left to create additional lift for the heavy engine which was positioned to the right of the pilot.



Front view of the 1903 Flyer I (picture source). Note the offset engine position.