It sounds like a motor fighting to find magnetic center. Traditionally, this is believed to be caused by an incorrectly set distance between shaft ends (DBSE). And when we see this, that is often the case. But, in some cases, we seem to find that DBSE is not the root cause. If the coupling is a disk-pack style, it will have a certain spring rate and an axial natural frequency. I think that some machines tend to excite the natural frequency of the coupling in an axial oscillation mode. If the pump shaft is not moving in the same manner, it is probably because the pump has rolling element bearings with much lower axial clearances, or it has a strong and stable thrust that resists this motion better than the axially soft motor rotor.
But, start with the easy answers and work to the harder ones. Verify that you have a good alignment in hot, running condition s. Verify that the DBSE is set correctly including allowance for axial growth of the running shafts. A certain about of cold pre-stretch may be needed to achieve a proper DBSE on the running machine train.
Johnny Pellin