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The codes you list indicate a major problem with the hybrid battery pack. The usually mean the battery needs to be replaced. They could also indicate a bad hybrid ECU. How much diagnosis/repair do you plan to do yourself?
Do you have access to a Techstream totoya scan tool?
A Techstream is about $2000 used and will likely not have the latest Toyota updates since they are around $1000 per year as well.
Some of the functions can be performed with a Snap on Solus or equivalent but you won't be able to do everything a Techstream will.
The C1259 and C1310 usually only set with a communication fault between the regen braking module and the HV battery. So when you read that c1259 and c1310 set whenever there is a no start condition that is probably backwards. The no start condition is likely a communication problem with one or more other modules, which in turn sets the c1259 and c1310.
The C1378 indicates a voltage supply problem to the brake control system.
The brake control power supply is in the right side of the trunk by the 12 volt battery. At the brake control power supply connector B18 pin 8 Blue(L) wire, check for 12 volt battery voltage.At B18 pin 5 Black wire, check for 12 volts with the ready light on.Check B18 pin 14 White/Black wire for ground.Verify that at B18 pin 12, a Black wire and pin 13 Pink wire, there is a 12 volt square waveform when the ready light is on.
I think you will find a voltage supply issue, a bad connector, or a blown fuse somewhere. It's just a matter of finding where it is.
This diagram shows how to run the test:
I'm confused about where we sit on this. The gas engine won't start and the electric motors won't run, but there are no trouble codes? The post you attached about the identical problem lists several trouble codes and a number of battery checks and disconnects performed by the owner. You don't have any trouble codes and have yet to check the individual cells of the HV battery right?
You are telling me that you are having the exact same problem as the people in the forum links you keep sending me. They have trouble codes- you do not. They have attempted multiple checks on the hybrid battery- you have not. It seems to me that the problem you have has only slight similarities to the chat you sent in that it won't start.
Your question has dropped to my timed out list.