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Your rising hot air balloon is not in free-fall. Free-fall means, that there are no other influences acting on the object then gravitational. So to determine the weight of the hot-air balloon with the "Bartlett" definition we need to evacuate the atomosphere, or at least the local region where you are determining the weight force on the balloon, measure its free fall acceleration in the desired frame of reference and then compute m*g_free-fall.
So I conclude that it is not problematic but is well-defined.