I think it's best to start by pulling up the datasheet of the batteries you are using. For example, here are some Energizer AA and AAA alkaline datasheets:
https://data.energizer.com/pdfs/e91.pdfhttps://data.energizer.com/pdfs/e92.pdfThe lowest published capacity and current draw data point for both is 25mA, so ~3,000mAh for AA and ~1,150mAh for AAA. Even though both DMMs draw less than half that current, given the low-voltage cutoffs of the DMMs I think the numbers are probably reasonable as-is for this theoretical exercise. Some of the batteries' capacity will be unusable.
The BM78x has 3xAAA batteries, and the 121GW has 4xAA. The mA draw for each that is published is based on the operating voltage of the DMMs, and we know both are wired to run the batteries in series, so for runtime calculations you would simply use the mAh capacity of one battery.
One variable is that devices typically require a specific amount of power (V*A) to run, so as the voltage drops, the current will increase to compensate, and vice-versa. For the 121GW it might be 30mW (4*1.5V*5mA) and for the BM78x it might be 36mW (3*1.5V*8mA). We have to make some assumptions.
So for a true number, you would have to characterize the DMM with a power supply or do real-world tests with actual batteries. Another variable is the point at which each DMM begins to become inaccurate, even though it is still "operating".
However, for back of the envelope calculations, I think we can start with:
121GW - 3,000mAh/5mA=600 hours maximum
BM78x - 1,150mAh/8mA=143.75 hours maximum
The BM78x manual states the low battery indicator kicks in at 3.7V, which is 1.23V per cell. That is leaving a lot on the table. So I would lean towards ~100 hours as being more realistic.
The 121GW has a published life of 500 hours typical, and that seems like a reasonable number at face value.