Does my Soldier have to show his temporary profile to me as his supervisor? If not, is there a regulation governing that?

I was had the understanding that a soldier had to carry a copy on his person at all times and be ready to present it when needed. I know I was a squad leader for a WTU at Ft. Hood and all 16 of my soldiers had to carry theirs.

While in command and having frank discussions with my lawyer I was amazed at what Soldiers could get away with. If you asked in good faith, and the Soldier fails to comply I would elevate it. However, there is a need to know element and individuals have a lot of protections they do not realize. As an example you do not have to cooperate with and investigation. You can stall, no show, and annoy the investigating officer for quite some time. I wouldn't recommend it though. However, the commander can put a stop to it and give you an order and if you fail to show for that he can get you for violating his lawful order. The UCMJ has a lot of nuances, and the commander had trial counsel to advise of what the left and right limits are.

I think you will find a barracks lawyer that says no, and a Jag lawyer says it depends.
If there is a need, based on a physical activity required for that is otherwise restricted and the person asking has a direct supervisory position, then yes, and Id bet a JAG lawyer will support it. Though I've never had the issue so never had to ask.

But remember I bet the JAG answer will depend on how the question is asked.
PSG to SM ...Sm back from sickcall,,, "Let me see your profile" Id bet the strict answer is not required. though I've asked that question about a million times and never had an issue.
PSG to SM after sickcall... "go to the MP and help the crew bust track" "PSG, I cant it violates my profile I got this morning:' "Ok, let me see it please " Now Id bet the JAG answer is Yes, supportable and legal"
AR 40-501 may have more specific info
http://www.apd.army.mil/epubs/DR_pubs/DR_a/pdf/web/AR40-501_WEB_Final.pdf

I had a Marine get in a decently heated argument over this same subject. What it ended up as is that as a supervisor you are authorized to be informed of the limitations and duration placed on the Service Member. Only the Commanding Officer has perview to the diagnosis and treatment plan and even then there are limitations as to why they are told and only by the medical officer.

In our case our compromise was a signed redacted copy for his training jacket.

Yes

Yea under the HIPAA law that soldier does not have to show you his profile even if you are his team leader/squad leader/ply sgt or plt ldr. Yea he does have to carry it on themselves at all times but the only ones who can truly ask to see it and not break any rules under HIPPA is the Company C.O. and Company 1SG

If acting in the scope of your duties, if you were to say "let me see a copy of your profile", could that not be construed as a lawful order?

I would certainly want to see it, to ensure the Soldier and everyone else is understanding the profile and not violate it.