318 runs real load and shakes like crazy

I’m helping my father restore a ‘70 Duster with 318 motor. We initially could not get the engine to spark and run. After replacing the rich, charred, plugs with ngk basic copper ones and cleaning the starter relay connections.
Sounds like you got a used engine and know nothing about it's history. yes?
So you changed plugs and tried to start it. yes?
And then you succeeded.
The car runs really, and I mean REALLY, loud.
Are you talking about the engine clacking or are you talking about exhaust?
Can't tell from the recorded audio. Capturing sound, especially loud sounds, is an art and science. Just tell us what you mean.
The timing can’t seem to be dialed-in; as the notched line in the harmonic balancer isn’t showing (at all) with the three timing guns we have.
That's probably really straight forward. The timing is beyond the range of the factory marks.
I'll go further and say its probably too far advanced to be read at whatever rpm the engine was running.

Lets say this is factory engine with a distributor from 1969.
The dashed gray line represents the advance curve measured for this specific distributor.
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I think the factory timing marks only go to 10* Before Top Dead Center.
So with this distributor, timing above 1100 rpm will not line up with any factory marks.

Looking at your engine it looks like its sporting a Street Dominator intake and some other bits that suggest its been hot rodded.
In which case it will want timing and advance closer to a non-smogged 273 4bbl, or an older Mopar Performance distributor.
Instead of 0 or 5* @ 650 or 700 rpm, it would want 10 or 12* at 650 to 750 rpm.
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The more radical the cam, and the lower the compression, the more initial it will want.
But the point is that initial of 10 - 12* is at the edge of the factory marks.
Cheap answer is timing tape. Get one with a dark background.

I’ve tried setting the curb idle and carb set screws. Fully closed set screws and /or fully open set screws currently have zero impact. If I adjust curb idle (barely lower) the engine stalls out.
Yup.
Classic indicator of hot rodded engine.
It will take several iterations of increasing timing and reducing the throttle position, then readjusting mix.
The idle mix is only the fine adjusment on the idle circuit.
The transfer slots also provide a portion of the fuel at idle. Open the throttles too much and the transfer slot contribution overwhelms the idle port contribution.
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Maybe obvious but the engine must be warmed up before it will idle at low rpm.