I haven't seen anyone mention eye dominance.
Check that you are right-eye dominant and not left-eye dominant. That could easily be the problem. Also, if you have a buddy with a video camera, have him
set it up on the bench in front of you and flip around the screen so that you can check the focus. Focus on your face and you'll be able to figure out if you've got a flinch preceding the shoot.
Rent a revolver, have your friend load only four chambers, and randomly spin it so that when you get it, you don't know which shot will fire. This is called the ball-and-dummy drill. If you cannot spot a flinch or other flaw in your method from this, I don't know what will.
If the range owner continues to object - especially to this drill - find another range. I have yet to hear ANY good reason why dry fire is a bad thing - other than possibly hurting the firearm.
Some pistols can be safely dry-fired. If that's the owner's objection, ask for a revolver that can be. Alternatively, buy some snap caps and load the chambers with some live rounds and some snap caps. That should address his concerns.
Hope this helps.. let us know.