This doesn't seem right at all.
The bearing arrangement is essentially that drawn on page 2 of the document you can download here:
http://www.model-engineer.co.uk/news/article/lathe-bearings/19136
If you have longitudinal movement, then you should be able to push the spindle to the left and take up the slack with the c-spanner.
If you can't take up play by adjusting the bearing then either the bearing is stuck OR have assembled it incorrectly so the c-nuts are not contacting the bearing.
Leather mallets should not be needed.
If the bearings are not seated properly, they will move as soon as you use the lathe in anger., Stop, readjust the preload and continue…
The hard plastic spacer is more than man enough to apply preload to the roller bearings, I think they actually need rather less than the original ball races. Shortening it is not a big issue, it needs to be about 1cm shorter to make sure the bull wheel is correctly aligned. At a push you could cut it with a junior hacksaw and finish it by hand.
Neil