2013 Prius C, hybrid battery/Yaris?

No. Without the traction battery, the car won't start.

 

Same size engine, but not the same engine. OK, it shares many parts with the Yaris engine, but the big difference is that the Yaris engine is Otto-cycle (high compression) where the Prius is Atkinson cycle, low compression. So the Prius version burns less fuel & makes fewer horsepower.

But the hybrid transmission is sort of the trap- it requires both gas and electric systems to be operational to drive the car anything like normal.

There's good news, traction battery prices recently came down.

 

All depends on when you sell it. Time it right and you get out while it's good, get good dollars out and the new owner gets some fair miles before they have to do anything.

Plus there's something to be said for the smoother transmission you get to enjoy for X years vs. the 90s-tech Yaris 4-speed. The Yaris might have been the cheaper choice overall, but you've got the better car for now.

 

Depends. It's around an 8k difference (new) between a Yaris and a PC. The hybrid battery seems to be going 150k + miles from the average-ish that I've seen. A new battery can be had for less than $3,500 installed at the dealership. It's even less if you can do it yourself.
Hometown Hybrids in Texas will do it for $2400 installed.
In my 115k miles, I'm averaging 50mpg. I've spent ~ $6000 on just gas.
According to the calculator on fueleconomy.gov , I've saved $3,565 over a Yaris. So essentially, it's a wash after buying a battery if I had to buy it today.

 

Is there a way around the whole traction battery thing or must the hybrid battery be operational to engage the traction battery in the car?

 

No, you need both components for the car to work. The hybrid battery also provides around 24hp, which you need when your max hp is 99.

You can get brand new blocks like these (currently sold out, though) for less than $1500, and if you’re handy and/or with a little help you can replace everything yourself.. which means $0 for labor.

With the cost of used cars and gas prices going through the roof right now, the Prius C remains one of the most fuel efficient and reliable hybrids you can get in the North America, and IMO $1500 isn’t an astronomical price to pay if it breathes new life into your vehicle.

ETA: I wasn’t paying close attention to the different “A” names and wrote this response thinking you were the OP coming back to ask the same question 2.5 years later.

 

Any Prius has no 12 volt starter motor, it uses the High Voltage battery to run the AC Motor/Generator1 to start the engine.

Additionally, only Motor/Generator2 pushes directly against the tires, M/G1 and the engine only assist M/G2, never push without M/G2, so you have to have the High Voltage battery.