I am not familiar with this model. But I am picturing a double suction, single stage, axial split pump. If so, I would suspect a hydraulic problem that is causing axial shuttling. If there are any elbows in the suction piping that turn the line parallel to the pump shaft, this could be a cause. Even a gate valve with the valve stem parallel to the pump shaft can throw uneven turbulence into the two impeller eyes. If the pump is running at too low a flow relative to the best efficiency point, it could be experiencing suction or discharge recirculation that is driving the shuttling.
We see this axial movement regularly on pumps of this configuration. I am often amazed at how little movement we are able to get when we shut the pump down and thrust it with a dial indicator. A very small amount of axial movement on a running pump looks like a lot more than it is.
I would review the bearing selection and arrangement. Make sure that the bearing is properly constrained from axial movement. For a back-to-back 40 degree angular contact thrust bearing pair, we would normally limit the outer race movement axially to about 0.005 inch in a large between bearings pump. Review the piping configuration on the suction side for anything that would throw uneven flow to the impeller eyes. Review the operating conditions in light of the suction specific speed to make sure you are running at an acceptable rate relative to BEP. Of course, you should consult with the pump manufacturer. They may have experience with this particular model that explains the problem.
In the worse case scenario, we have resorted to changing the impeller eye wear ring geometry slightly to induce a stable thrust in one direction. But, this would have to be done with extreme care.
Johnny Pellin