Put trained animals in it, probably three bats connected to a hydrophone system. Train them tug a switch when they detect a certain sound profile. If two or more bats trigger at the same time, it arms the mine. Bats can process audio signals to a staggering degree, better in some cases than modern digital systems.
B.F. Skinner famously trained pigeons to guide arial bombs but the end of war and electronic technology made the weapon obsolete before it could be deployed.
I don't think there are any fish that use accoustics but the lateral line sense organs on all fish can detect patterns of vibration. The mine could have a hydrophone that picks up ship noise, amplifies it and then plays it back through an underwater speaker to some trained fish. Whose movement would arm the mine. The trick here would be training the fish, because their behavior is fairly inflexible.
Mammals or birds are your best bet. Bats are actually heterotherms i.e. instead of having one set body temperature, they have several they can pop back and forth between. They can be dropped into hibernation and then woken up, all within a few minutes. See the bat bomb experiments from WWII. Some species of bats will also go into hibernation if the O2 levels drop to a certain range, then wake up if they get to low. Your mine would likely be a combination of a simple noise detector tuned to metallic or general prop sounds, which would wake up the bats so they could judge whether the sound represented a valid target or not. If not, it would put them back in hibernation.
A few pounds of oxygen generating and CO2 absorbing chemicals could keep three mouse size bats alive for several months, even without hibernation. They wouldn't need much food and none if they spent most of their time hibernating. Cooling the bats could be done with any number of endothermic chemical reactions that don't produce a gas.
I suggest animals because the analog circuit of the era could not process different sonar profiles without rebuilding the circuit and they were easily confused ambient noise either natural of man made. The Allies easily spoofed advanced german noise seeking torpedoes by dragging two hollow metal cylinders behind the ship that banged randomly from wake turbulence.
The extra gear to keep bats alive for a few months, dipping in and out of hibernation, won't be any more complex than power requirements of analog circuits of the era. It's not particularly cruel to the bats either. If the mine gets lost, they just drift off from hypoxia, if the mine detonates, they never feel a thing.