Code P1386, "Hybrid Battery Low", Check Engine Light on and a noticeable loss of power?

The hybrid battery is charged by driving the car, can you get the engine to crank over? Is the battery you installed new? Maybe we should see if the engine is locked up, let's use a breaker bar on the front of the engine to see if it will turn. Check the diagrams below to show you where to install the socket for the breaker bar. The P1386 code shouldn't make the engine not start. The socket size should be 21mm but that can vary. Check out the diagrams (below). Let us know what happens and please upload pictures or videos of the problem.

It turns over and I can feel good compression. I would normally pursue a possible starter problem. The thing that confuses me is the Dealership saying the engine needs to be replaced. Is there a fuse or relay for the starter?

That's a good sign. It sounds like the regular 12-volt battery might be bad because the engine should start. Please locate the battery to check its condition. This guide can help with the diagrams below. Also, you can use a voltmeter to check for 12 volts at the battery.

https://www.2carpros.com/articles/car-battery-load-test

and

https://www.2carpros.com/articles/how-to-use-a-voltmeter

It looks like the alternator is the starter, check out the diagrams below. Please don't touch any of the high voltage lines. They are in orange. Please let us know what you find. We are interested to see what it is.

I'm stumped. The car runs great when it's cold. But as soon as it warms up, it starts skipping occasionally at low and medium RPM's. If I accelerate, it goes away (at least temporarily. There are no codes on the ECM. I like the car but I can't handle the skipping. It's not rhythmic so I don't think it's a plug or coil. I replaced the Variable Timing solenoids, all coils and plugs, the Mass Air Flow sensor and a few other items. But nothing helps.

Got any ideas?

This sounds like it could be the MAF sensor is off but not enough to trigger a code or the fuel pump is getting weak. Here is a guide to help you test the pressure you get rent a gauge from the auto parts store for cheap. The pressure should be:

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

Fuel Pressure.

Start the engine and measure the fuel pressure at idle.

Fuel Pressure .................... 333 - 353 kPa (3.4 - 3.6 kgf/cm2, 48.3 - 51.2 psi)

If low here is how to change out the pump in the images below. Check out the images (below). Let us know what happens and please upload pictures or videos of the problem.