I dont think you would have any problem with that. But lets make sure I am understanding what your saying.
So basically you will lose less than a 1/4" off the top of your tapper?
If I have that right again I dont see any problem with runout, or holding ability. Now with it being open up there there may be a bit of the collect sticking out of the top of the tapper ( which I guess you want it to).
So you must be wanting to add a pull stud to the collets and need room for the pull stud accuator ( pull rod)?
One thing I can think off is what will be left of the spindles wall thickness? Or will the rest of the spindle even be drilled ( only top of tapper needs drilled out?)
The edge of the top of the tapper should be debured and maybe even a ever so small of a radius so the colect doesnt try to grab the edge when you pull it out. Or maybe even cause a line to wear on the collet if its not dressed/debured.
I would also take a very good collet and blue it and check for contact of the spindle tapper to make sure everything is good before you even cut it. Then do the same after modification. Pull bar/draw bar should have the best alignment possible so it doesnt try to effect collet seating.
Now I have never done this before and its just my opinion on the subject so you might want to be sure and get others thoughts on it to. But if the spindle is not super high dollar and could be replaced without much problem I wouldnt hassitate.