Do the Lee sizing dies in .40 have a different taper at the mouth than Hornady carbide dies and would the Lee die allow the cases to enter the die more easily?
Lee dies are no better, as far as the chamfer/fillet at the mouth of the die.
Some RCBS dies do have a wider mouth, but the chamfer is a shallower angle. So, it doesn't help much. (And, I don't know about .40 S&W.)
Dillon dies are substantially better, but I don't know how well they work on the LNL AP.
If you're having problems with case alignment and
primer seating, I would take a close look at indexing. I don't have much experience on the LNL AP, so I can't be more specific.