Could you aim the laser beam onto a mirror, set at an angle, to incresase the length from laser to target?
Although you seem reluctant to modify the machine, fixing the radial position of the hed would be an improvement to a machine which is a Mill / Drill (ie a beefed up pillar drill)
If a bracket is fixed to the head, so that a reamed hole in it runs on a vertical rod, fixed at top and bottom of the column,, this should satisfy your need.
Cromwell Tools will sell you a 1 metre length of silver steel, for the guide rod At least 1/2″ diamedter would be ny choice)
Drill and tap the rod, once cut to length, to secure to the upper and lower brackets. The hole in the bracket foxed to the head should be resaonably long, and a really snug sliding fit on the silver steel rod.
My method would be to fix one bracket to the column, fix the bracket to the head, and then to feed tghe alignment bar thraough it and secure to the bracket. The head should then be adjusted to the other end of the travel, a so that the alignment bar is vertical, before drfilling and tapping for the fixed bracket.
In this way, the head shouild be kept in alignment as it is moved up and down the column.
Obviously, the brackets should be as short and sturdy as possible, to maximise rigidity.
The problem might be finding the locations for the brackets!
Howard