You definitely need a watchmaker experienced with Omegas and almost certainly one able to get hold of movement data and parts.
Have a search through watchguy.co.uk and look for the Heuer Autavia he dealt with with similar symptoms.
Can understand in a way why your own watchmaker isn't willing to take it apart and look for things like the wrong wheel/pinion having been replaced somewhere. Tedious job and a lot of arithmetic !
The timing machine only checks the oscillations of the balance wheel, not the movement of the hands, so something is amiss between the two.
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