Does your second fob, do the same when placed "in the kitchen" in the same position? Wild suggestion that costs nothing, but refresh the batteries and then put the two fobs (you have two?) in i, the same place overnight and ii. different places over night - the dodgy one in whatever approximation of a Faraday cage/ electrically silent area you have.
I agree it's a bit of a conundrum that one battery should discharge more than the other. and such a short space of time - maybe check the battery mounting ie that the contacts to the battery stack are connecting either side of the 6v unit, and there is no intermediate contact to one battery only - is it always a . the upper or b. the lower battery that's being flattened?