Honestly the CF weight imbalance ain't that bad.
I run my Barracuda with 3.55s and a GVod so 65=2240 by the math. So I could be cruising down the road a few miles sometimes, before I even notice it. The only reason I know it's the flyweights is because a clutched-blip or three solves the problem. I mean at 2240 it is extremely subtle. Even when I was running 4.30s one summer, 65=2710 , it was only occasionally more than subtle.
The shake of OP's engine is not solely from the CF-flyweights; it may be a contributor, but at idle it would be extremely small . And anyways, my CF vibration never cycles in and out, it would be there 100% the same intensity, with no oscillation.
I think the flyweight idea is a dead-end..... at idle.
OP
After reviewing my memory banks, I'm concluding that the principle source is the crank. I say this because I remembered a certain 360 that came in one day and I couldn't find the source and it was the worst at idle. But there was no oscillation like yours. Since I didn't know what it was, I told the customer my opinion was a crank imbalance and recommended to the customer to get a second opinion; or a third even. A couple of weeks later the customer showed up with a big grin on his face, and the solution had indeed been to balance the crank.
But I am pretty sure that after the balance job, you may find a different secondary vibration to solve. Have the crank checked for straightness,etc while it's out and before the balance work.
That's my opinion.
However;
It may be as simple as;
the TC is of the wrong balance, or
the front damper is wrong.
Or both.
Before I would take the engine apart, I would look into that.
