Chosen Solution
As they get older, a FItbit Aria scale can drain four AA batteries in a week. Batteries should last months. This is apparently a common problem, possibly due to strain gauge sensors becoming flaky over time. Has anyone found a fix?
With further research, I found that the Aria has a poorly designed activation mechanism in the feet, with a plastic flex part that breaks, causing the activation contacts to randomly short, which keeps turning on the display. This happens repeatedly from any vibration or temperature change, and drains the batteries. The solution is to repair the flex part, possibly with glue or resin, or 3d print the replacement part from thingverse.