Sounds as if the taper is pretty worn and/or there are mechanical clearance problems with the pinion/nut.
If the nut simply bottoms on the thread before the taper is properly engaging, and if there is clearance between pinion and timing case with it tapped home, then might you be able to sort it with a thicker washer under the nut (or if it's an ATD, with a thicker horseshoe washer on the outboard end)?
In any event it's a bad idea to tighten a 3/8 BSF nut against the little 3BA thread and small brass key on the cb assembly - it's an unfair contest with only one winner!
A useful wheeze (I find, anyway) is to remove the earth brush from the mag casing and screw in a hard plastic or soft brass plug (5/16th BSF) until it contacts the armature. This should hold things exactly where you want them while you get the pinion into place and starting to grip.
If you don't have another decent contact breaker backplate, you might be able to reclaim the scrunched keyway with a home-made 'broach', or you could make it good with solder and Swiss files (painstaking but it will work). Or you could make another keyway exactly 180° opposite the damaged one. Its position is very important, but it is only a register and it shouldn't be doing any real work.