It does sound like the car is going into a limp mode. My limited experience is that limp mode is cleared when the car is turned off and does not activate again until the car reaches the condition of RPM, load, and temperature that triggers limp mode again. However, it sounds like on this car, limp mode is cancelled only by resetting the PCM.
You are correct that an obstructed exhaust would still be obstructed after a PCM reset, but it may take a certain set of conditions for the PCM to conclude “Wait a minute - this combination of RPM and load and throttle opening and manifold pressure and mass air flow don’t fit together - something is very wrong!” and it goes into limp mode…
Though the exhaust may be obstructed, that would not make the car shake and idle rough. For that I would look for the cause of that camshaft timing error code. A common failure point on cars is the camshaft position sensor, but that has its own code so it seems unlikely that it is the problem here.
At 7 years and 68k miles, you are out of the Infinity warrantee period. There is a remote chance that you might get some help from the Federal Emission Performance and Defect Warranties – 96 months/80,000 miles, but with modifications done to the car, I would bet against it. In addition, the modifications you have discovered so far may not be the only ones. Sorry.
It could be worse. When I was your age, I bought a Vega.
We all get bitten sometimes on used cars. When I was 60 years old, I wanted a white BMW wagon with a stick shift. I finally found one with 100k miles on it. I paid $10k. I spent 6k and many hours over the following two years on that car. The good news is that I still have it, now with 252k miles on it, so the /mi did not turn out to be too bad.