One of my Prius won't start the engine....

Electrically. (Or both in a way, electrically isolated by the mechanics of an electrical relay.)

 

Even, one could say, electromechanically.

 

I'm trying to think of a case where actual mechanical isolation is controlled using electromechanical means. The only thing that immediately come to mind are release/capture mechanisms (solenoid controlling a lock or latch for instance), scaling up to docking clamps (albeit mostly in sci fi). I guess a robotic "hand" would be in this category too.

 

Two batteries high voltage batteries will likely boost the odds of one of them being good but you won't know until they are charged and balanced. Maybe you can find someone near you to borrow the equipment from if you don't want to buy the gear?

 

The temperature of the battery/car might make a difference, if the car is just at that "can do it only under the most optimal conditions" stage. I'm guessing that freezing cold (as if that is possible this time of year) would be the worst choice. What would be ideal? Looking at the plot here for the battery mostly discharged (closer to 100):

Nickel Metal Hydride (NiMH) - Euro Energy guide to battery chemistries number 1 - Euro Energy Resources Limited

it looks like 45C would have the most current, which may or may not be enough current. It would also make the viscosity of the oil in the motor lower. Don't go above that (assuming that is even possible) because the trace for 60C is very close to that for 0C or even lower. Note that the temperature below the hatch floor can be considerably cooler than in the cabin. If the day isn't very hot retract the hatch cover and pull out the floor, and the plastic tray, then allow time for the pack to come up to temperature. It will lag the inside of the car by quite a bit.

Honestly though, I'm thinking this pack will need to be charged, and most likely also balanced.