Is this really unfixable?

since you're running an 05... you might also want to take in account other preventive maintenance.. such as cleaning the HV battery fan. it turns out that 5 years down the line they do kinda get clogged up.

the only other time i've had funny issues similar is when one of my ignition coils failed. yours doesn't sound like that but it's something you may want to keep in the back of your mind.

another preventive items. water pump and inverter pump as semi mentioned above. the water pump will make a lot of noise when it starts to fail... the inverter pump has recall campaign (i think it's the inverter pump)... with the car Ready (fully on) you'll see turbulence in the reservoir above the inverter. you may have to pop the top to see. it should be flowing anytime the car is on.

make sure your engine air filter is clean... you can't clean it, you have to buy a new one.

make sure your oil isn't overfilled. it's only 3.5 quarts.

(i think that's all the general things that you can look out for)

btw, how many miles do you have? maybe it's time to change fluids?