Has anyone had these odd electrical symptoms? 12V battery keeps shorting out each use

Anything below 12.7 is in need of a charge or replacement and most people on PriusChat want you to measure the 12.7v at the front of the car at the jump points which won't be as high as the battery. Also that 13.9 number should be 14.1v...
Check your connections, something is loose.

 

So the car has a bad negative connection between the 12v battery and chassis ground OR the positive connection from the 12v battery to the front fuse box has a problem.

 

ok, so you understand that the heavy red cable that is connected to the fusebox at the jump point is the same heavy red cable that plugs into the fusible link assembly on the positive terminal of the battery? For whatever reason, you are not getting continuity between the 12v battery and the car. You have a (+) cable, a fusible link assemby on the (+) terminal, a 12v battery, a (-) cable.

One of those items is not doing it's job.

 

I had the same problem and used the opening and shutting of the door to fix it for a little while, but it was fixed for good when I took apart and cleaned and inspected and upgraded all the connections around the 12v.

 

That confirms that there is a bad connection between the 12v battery and the front fuse box. It could be the negative or positive connections.

 

Charger yes, high amp no. At least some of the 12V batteries have a sticker that says not to charge them at more than 4.x volts. (I don't recall what x is.)

Just for the sake if dotting the i's and crossing the t's the OP isn't seeing the dashboard capacitor issue, right?

 

With the car off start at the battery and measure the voltage between the two posts. It was 12.6 V at least once. Now start moving one probe or the other outward (electrically) from the battery. Move the negative first to the other end of the cable, then to a piece of bare metal, like a bolt head nearby. Then go in the other direction. Back probe the heavy 12V line that goes to the fuse box. Then measure at the jump point in the fuse box. In all such cases it should still be 12.6 volts between positive and ground.

I also suggest unplugging the 12V cable where it goes in near the positive post and inspect both the cable and the socket it plugs into for damage. If that was not seated properly at some point it might have arced, which can make a mess of later connections. With that cable out measure the resistance from the positive post to the conductors in that socket. If the fusible link has blown that will be high.

Pull the 12V battery and attach it to an old fashioned load tester, pull 20A out of it and verify the voltage hasn't crashed to something below 10 volts. The battery in my car only has a little over 10Ah capacity left, and even it can maintain 11.8V when 20A is drawn from it. (Note, electronic battery testers all say our battery is fine. It is not. A discharge test definitively shows that it only has about 1/4 of the capacity it should.)

 

Have you actually verified the heavy red cable is properly plugged into the fusible link assembly at the 12v battery (+) terminal? Maybe try to unplug it, inspect the contact surfaces and plug it back in firmly until it locks in place?

Have you tried using the jump pack at the battery itself? If you clamp it to the terminals and everything works, then you pretty much eliminate everything except the battery itself and the connection between the terminals and posts. Just be certain to not reverse polarity, or you'll smoke the underhood fusible link assembly and some fuses.

....and it is possible for a battery to have an internal mechanical busbar failure that allows full 12.x volts to be read at the terminals, but drops to zero as soon as a load is applied. Anyone with a reasonable amount of electrical experience should be able to help you solve this in 5 minutes or less.