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Ok... were jumping around definitions here... the 3rd observer would provide the axis, to intersect the source and receiver line of light, thus allowing rotation, thus allowing you to apply your tangental calculations.
Your definition of the experiment, has added a 3rd "chopper cam" top down observer, external to the experiment....
How does the closed frame, with all components on the turntable... know it's turning?
How does the closed frame know where it's center is?
The only way to get a source and receiver to rotate, so that you may apply your tangental mathematics, is the addition of a 3rd observer, providing the axis.
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Incomprehensible.
The correct way is to measure the centrifugal force at two points in the rotating frame. Since the centrifugal force is directed radially outwards, the intersection of the two radial lines will mark the centre.
Once the centre is known the radii at the points can be measured.
The tangential velocity is calculated by the formula
Tangential velocity = ((Centrifugal force X radius)/mass)^0.5
Regards
Steven