340 Engine Shake
I can see if I can track down a carb. How does the distributor react being locked out? So the weights and springs have been removed, so the efi can control the timing
One more question: how much idle time do you have on it so far?
Do this, go into 'large gauges' and set it to display the following:
AFR Trim
AFR Learn
IAC steps
Spark adv (I think this is the right name)
Let it warm up (above 170F) and let us know what you see for these values.
There are max limits to the learn and trim. Some engines wind up maxing out for various reasons, and that can cause issues with the system's ability to adapt at idle. There are ways to change the limits, or to even alter the base fuel so that it doesn't need to run min or max fuel trim/learn.
Spark adv should jump around a bit while it tries to stabilize the idle. You might find that the 'shake' happens when the advance increases or decreases. If so, it might make sense to increase or decrease idle timing a bit to help it smooth out. But until you can correlate which way the timing goes when the issue presents, it's hard to make a suggestion.
What are you running for fans?
Power steering?
A/C?
If any of those start drawing power, it could drag the engine down and cause issues too... something to consider.