The trouble with pushing a rod held in the chuck is it's unclear what's being measured. It might show the bearings are loose, but maybe not. It's quite easy to be misled, causing one to break the golden rule: 'if it ain't bust, don't fix it!'
I'm in favour of starting with 'does the lathe cut metal OK' before moving on to DTI's and so forth! Are there any signs the lathe isn't cutting correctly? If so what are they? The symptoms may suggest causes other than the bearings. And if the cause isn't obvious, it may be necessary to pin it down by carefully eliminating possibilities one by one.
Dave