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.

It is a mute point, you are out of uniform either way. You are supposed to wear your rank, what is the issue?

Never heard of that one, but unless that's the uniform stated by your supervisor you're outta uniform.
When I was a PFC a major asked (after I called everyone to attention and saluted) why a bunch of Privates were cleaning weapons in our jungle pants without shirts or Ts.
I responded with, "Me Sir. I told them we were too dirty to sit around in the sweaty Ts so to strip'm off."
He nodded and told us to carry-on.
Of course it could have gone another direction.

I worked for a 1Sgt. that would tell you if you don't want to wear all of your rank take it off and I will adjust your records to reflect that rank.

Rank insignia on a uniform should be consistent and not mismatched. So, if you are not prepared to update the PC, neither should you update the blouse. Walking around with conflicting rank insignia is "ate" up.

At what point do you not consider this question to be common sense your leadership is setting you up for sucess.

From the first day in Uniform we are taught uniformity and how to properly wear our uniforms.

Please understand that leadership is looking out for you. Trying to buck the system or trying to find the exception to the rule will only set you back.

Let's practice critical thinking a simple look around would have told you the answer to this question.