David, can I chip in. Back in 2003/4 Bryan Mumford and I had a discussion about the "balance" wheel on the Eureka which was developed into a theory I presented at a UK AHS meeting regarding the screws and the wheel. What Bryan and I agreed was that the big wheel was not a balance wheel, and that the adjustment of poise was deliberately designed into the clock to allow for regulating.
My Eureka has large balance screws with three different weights (picture). Swapping light for heavy between the top (Balance wheel at the point of impulse) and bottom changes the rate significantly. Decreasing the weight at the bottom slows the clock down (and increases the amplitude). Decreasing the weight at the top (decreases the amplitude) and speeds it up.
In a "proper" clock
-----Is controlled by the escapement and provides just
| enough power to overcome the losses in the escapement.
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The Primary Power Source (Great Wheel)
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-----Also drives the hands (Time Train and Motion Work)
OK so the first couple of wheels may be in both paths, and the second hand may be driven from somewhere in same train as the escapement. But the escape wheel is always on the end of a train. The makers always trying to achieve the theoretical "free escapement" to allow the balance wheel to operate with the least possible interference. An "Escapement" balance would always be poised as perfectly as possible to minimise it's deviation from a "free escapement".
In a Eureka on the other hand
The Primary Power Source ( the battery and coil)
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Drives the "Big Wheel" by magnetic impulse
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Which drives the Hands via the cam, roller and push click (for want of a better word).
These two processes (magnetic impulse and mechanical push) occur over different segments of the arc of the big wheel. This means that the electrical impulse has to impart sufficient energy to the wheel to both overcome losses and provide motive power to the motion work. Energy that is temporarily stored as kinetic energy in the Big Wheel. Given this "interference" the Big Wheel is never going to be anything like a "free balance".
On the forward swing the kinetic energy in the wheel created by the impulse transfers into three places (ignoring friction and other losses) - the spring, driving the motion work, and into changes in the potential energy of any imbalance in the weight of the balance wheel. If the top is light and the bottom heavy, the centre of mass rises and gravity assists the spring in absorbing the kinetic energy of the swing so the swing is shortened. If the top is heavy and the bottom light the spring has to absorb both the kinetic energy of the swing and the potential energy released by the centre of mass of the wheel falling - so lengthening the arc. In a perfect balance wheel this would not affect the rate because apart from frictional losses the same potential energy would be converted back into kinetic energy on the reverse swing. But in the Eureka the reverse swing has lost all the energy needed to drive the motion work. This difference in energy in the two swings seems to mean that the gravity component has a much greater influence. Short arcs increase the rate and make the clock gain, long arcs make the clock slow.
Bryan and I concluded that this was deliberate design (otherwise why does the wheel need such heavy and expensive bearings) , and that the manufacturers would probably have rated production clocks by setting up a standard pattern of screws, run the clocks for a given period, then there would have been a standard table of adjustment - (e.g if 5 minutes fast swap no 3 screw for a lighter one etc, etc, etc). This would have allowed them to employ semi-skilled production workers rather than skilled clockmakers to set up the clock.
For your clock I suggest weighing the screws and then swapping them around until you can get the clock to run fairly accurately with them all in. Heavy at the top will slow it down, heavy at the bottom will speed it up. Trying to keep the wheel poised is unnecessary and probably counterproductive.