'Jam the stock into a bearing'?
It sounds a good idea but a bit impractical in this instance. The corner to corner size of a 2" square is over 2-3/4" (71.27mm) so it would need to be quite a large bearing! A ballrace is likely to have a significantly larger OD (if the ID is not to be distorted by the stock corners), if we are just talking about a plain sleeve bearing that too would need to be quite chunky so its OD remained circular.
If the OP had a fixed steady large enough to accept the bearing OD he is likely to have a 4 jaw big enough to hold the bar for drilling the end.
With care I would face the central part of the the end before using the slocumb bit, then face the end whilst supported by the tailstock. Drilling, even a large hole, should not impose any side loads but its best if the gripped end has already been faced and is firmly in contact with the chuck face.
Ian