Using CAD modeling, the center of gravity of a rocket can be easily predicted. But in reality, after the assembly, the CG will not be exactly at the place given by the model because of difference in manufacturing of components and other assemblies like plumbing and wiring etc.
So, how is CG determined after complete assembly of the rocket?
Modern CAD software is very good and can include the mass of every zip tie and weld if it's made with enough care. If the real life doesn't match up with plans, the whole rocket can be put on multiple load sensors and from that the CG can be calculated. However, finding the CG is still important, especially in flight when mass is changing due to fuel consumption and rocket staging.
When an object is in flight or is not constrained, it rotates about its CG. This means if you spread accelerometers around the object and compare the different readings of those sensors, the CG can be calculated quickly. For more detail on this technique, you can read this paper: https://www.mdpi.com/1424-8220/14/9/17567/htm
For the Apollo program, there was a device that measured the weight and balance of the completed modules. Page 9-4 of the Apollo Program Summary Report describes the steps in manufacturing the Command Module:
9.1.1.3 Final operations: In the final phase of command module manufacturing, the vehicle was cycled through another tumble-and-clean operation in which the vehicle was rotated through 360° in each axis to dislodge and remove debris. The weight and center of gravity were then determined, and the vehicle was subjected to an integrated test (sec. 9.1.5). The command module was subsequently moved to the shipping area and prepared for shipment. Such items as crew couches and crew equipment were removed, packed, and shipped separately.
The manufacturing of the Service Module is described on page 9-6:
Following the tests, the vehicle was cycled through the tumble-and-clean positioner to dislodge and remove debris. The cleaned vehicle was then weighed and its center of gravity determined. On completion of these operations, the vehicle was placed in the integrated test stand for the integrated test series described in section 9.1.5. The integrated test completed the manufacturing, test, and checkout operations, and the vehicle was mounted on a shipping pallet and prepared for shipment.
Both operations were performed on a piece of equipment called the "weight and balance fixture":
The exact CG position is (never?) required
You just need a good CAD approximation (and maybe verifying the CG of complicated heavy parts if strong doubts still exist). Then you qualify your GNC algorithms taking into account all the variations you can predict (and some marging for the ones you cannot).
When accelerometers are used, for example on Ariane, it is to measure the dynamic state of the launcher, i.e. which dynamic structural mode is active, to avoid control errors (and amplifying dynamic mode)
