Where do I find the specific regulation that you cannot wear rank on your blouse if it's not on the patrol cap?

Your trolling us aren't you? Why would this condition exist? I can think of many times where one might need to remove their rank, so it follows that you would remove the rank from all pieces of the uniform that contain it. Are you planning on walking around without your rank?

Let me guess..you loss your pin on PFC from your PC and when your DS called you out he/she told you this. Not every single thing is written out word for word. There are implied parts. One of these is the idea that when you are serving duty you must do so in uniform, unless specifically authorized by the commander. The implied part is that the standard definition of the typical duty uniform includes some sort of "uniform" standard to your appearance. This would include wearing your rank in all the areas where it should be worn

That would be a local policy under AR " I'm the drill sergeant and said so". It is not in AR670-1 nor the DA Pam.

DA PAM 670-1 Page 165, 21-7 /3 a and b. Gives you this information don't miss read the insignia on the Head gear as optional only optional in latch and hook verses pin on. The only option their is to wear pin on insignia rather than latch and hook. One Other thing that may help if they tell you to wear it and in any specific way you may request from that leader the regulation requiring it however it is best to just follow what the leader says rather than try to come out as disobeying a lawful order.