Random misfires

You answered your question--the pistons have low compression. A good mechanic should be able to diagnose and tell you what the solution would cost, but it's doubtful it'll be cost-effective to repair. Is this a high-mileage engine that burns oil? It sounds like there might be good value in purchasing a low-mileage salvaged engine. Consider the health of the hybrid battery (over ten years old now?), the condition of the rest of the car, and its market value when making your decision. Your expensive catalytic converter may have issues soon, too, if the engine is in poor health.

 

These engines are well known to start burning some oil sometime after 150K miles, so it would be good idea to check the oil level frequently, say every other fill-up.

Is the engine running rough, and/or noisy?

Without records, you'll have to look at the battery. Some aftermarket battery cases are painted a bright color, easily seen after some minor disassembly in the luggage compartment. Rebuilds will have discontinuous serial numbers on the 28 battery modules, and you'll have to open the case to inspect that, a bit more involved. High voltage hazards exist, so be sure you know what you're doing there. The original modules will have a year code compatible with the model year--search this forum for codes if needed. None of this is worth it if the battery is working fine, and most are. I mention it as a possibly expensive repair in the coming years.

 

If the oil was ever run low, the bottom half of the engine could have seen some damage. That could account for noise and shaking. If it's more noticeable from under the car (be careful!), that's probably it.

 

It is normal for the car to shake from the engine during the initial warmup cycle, or when the engine is charging the battery at an idle stop, for example, when sitting in a parking lot with the AC on for a long time, and the battery drains to two pink bars.

If it is shaking when in neutral, or when accelerating, you have a genuine misfire problem.

 

What you describe is classic blown bottom end of a Prius. There's so much bottom end vibration its confuses the crank sensor and it will throw multiple misfires.
The car should never shake no matter what its doing at least mine doesn't. The rattle your hearing is the piston rod bearing.
Usually caused by running a car that eats oil low on oil. Throws a check engine light from oil starvation. Previous owner knows this and you got boned.

And assuming the traction battery has been replaced because the previous owner said so is silly.

Anyway go on youtube and look up PRIUS INSPECTION MODE and you'll be able to get the engine to run continuously. get under the front of the car under the engine while its running and listen for disturbing sounds like banging and thumping and vibration.

 

I think its time for a new engine, or used one.