After doing some more tinkering I believe I have discovered the cause of the problem and a fix for it. I decided to compare the primer arm and dispensor on my Turret press which work perfectly for me to the parts on the new press. I noticed that there is a slight difference in the height of the primer cup on the primer arm on the old vs new ones. Measured with a cheap calipers they are approximately .012" different in height.
If you place the primer arm into the safety prime head you can see that there is a considerable gap between the top of the cup and where the primer slides out. I believe this gap is too large and causes the primer to tip as it enters the cup. When the safety prime head retracts the flipped up primer is caught and gets knocked out of the cup.
So we need to raise the height of this cup in order to reduce the clearance so the primer does not have room to tip sideways. The cup is mounted on a steel pin that is press fit into the die cast primer arm. By sharpening a punch to a flat chisel point it was relatively easy to drive the punch between the pin and the dies cast body to force the shaft up and raise the height of the cup.
After raising the pin up out of the arm I set the height of it so that the cup is about .010" or so below where the primers slide out of the safety prime.
Next I tested this on the press by loading 100 primers into the tray and feeding them into the primer arm on the press, and then manual taking them. All 100 primers fed without issue and none were dropped. At first I was using the same slow and deliberate technique I had found to work yesterday, but for the last 50 I was just stabbing the primer are with one finger as I do on the turret and it still worked just fine being rather careless about it. I even tried lowering the press about 3/8" from the top of the stroke and they still fed fine.
Finally to prevent the shaft from sliding back in I gave it a quick blast with the mig welder into open window to hold it in place, and ground it back smooth. The primer arm is not steel so it will not weld, its a die cast alloy of some sort, so you need to weld directly onto the steel pin. If you just fill up that window with a good bead it will not be able to move.
Below you can see the heights of the classic cast primer arm, and the auto breech lock before and after modification.
I will relay these results to Lee customer service so that possibly they can do something about it.