Presumably the weight change you're looking to monitor mainly is volume of honey produced? Perhaps there's an analogue for that - maybe reflectivity of the frames or opacity to IR or some such?
Perhaps even just look at the heat retention against time accounting for external temperature (eg cooling rate at night vs. the ambient cooling curve) as that will give you the thermal capacity which should increase with increased honey volume. That should give you a first-order approximation at least (humidity would factor in to rule out waterlogging, not sure what the other main causes of hive weight change are beyond bee numbers which you'll also presumably have from the audio-volume and/or the ingress-egress readings).
Perhaps even just look at the heat retention against time accounting for external temperature (eg cooling rate at night vs. the ambient cooling curve) as that will give you the thermal capacity which should increase with increased honey volume. That should give you a first-order approximation at least (humidity would factor in to rule out waterlogging, not sure what the other main causes of hive weight change are beyond bee numbers which you'll also presumably have from the audio-volume and/or the ingress-egress readings).