I am being told to go to PT by the command on only 4-5 hours of sleep. As a junior enlisted, how do I solve this problem?

Do what your told.

The problem is you are in a medical unit. With all the different shifts. In a way it is like being in a M.P. unit that has different road shifts.
Most of them do there P.T. before or after their shifts, And that seems to work for them.
It is my opinion that your PSG should talk to the 1sg about the P.T. formation's.
I would see if the P.T. could be done before your shifts, Like most of the Army.

How do you solve the problem?
The first step is to identify the problem.
Your command wants you at PT. You are not at PT. Problem identified.
Get your ass to PT. Problem solved.

Is this a serious question?

Uhhhh...ya do it bud

Suck it up buttercup, this is the military not some cushy civilian gig. Get deployed to a hot zone and see what life is really like. Go do your PT and stop whining about it.

My feelings, you're a soldier first and you do what you need to show for unit formation. As a retired Health Admin type, I faced these issues all the time, you have freak working hours so go talk to your hospital first line supervisor and tell them of the situation and see if you can do a 1700 to 0030 and take a 1/2 hour lunch on PT days. This is my input but I'm going to be honest with you, I never got involved with issues like this with the troops, my senior enlisted and chiefs took care of it. An interesting talking point would be..... "Hey first Sgt, will you look at this? I get off at 0-dark thirty and race home to hit the rack, at the mid sleep hour mark they offer PT and at 1600 I'm scheduled to PT and work at the same time, can you help me figure out how to make this work? When I work my shift I give 110% runnin' and gunnin' all night, at the end of that shift I'm spent, I need my rest, I am afraid if I PT in a stuper, I may get hurt and then loose a troop to unit, PT and they loose me at the hospital. Everybody looses, I want to give you my best, can you work with me so I can do that?"

Um, go to PT and quit whining. You’re a soldier for crying out loud.

I was an Army photographer in West Germany from '71 to '73. I also, in my spare time, played in a band for USO/Special Services. We usually played only weekends, but occasionally we played Thursday, Friday, and Saturday evenings. We played in Em and NCO clubs from nearby or far away. On the far away gigs I would get back to the company area just in time to change into uniform and stand in morning formation ready to put in a full day's work. That meant I stayed awake from early Thursday morning until early Sunday morning with no sleep. I was young then and could do that. If you are off shift in time to do PT, that meant you had the time afterwards to get some sleep. I don't see what the problem is. Sorry.

Your section should obviously be made personally responsible for your PT. As a side note leadership would do well to assign the best examples of physical fitness available to them to this duty. An irregular duty obviously calls for utilizing small group tactics and increased responsibility expectations for the soldiers. Your problem seems to stem from non engaged leadership; my experience of this was mostly do to arbitrary prioritization at higher echelons within the chain of command or unrealistic expectations for key positions from the chain of command.

There are regulations dealing with sleep requirements. These specify doctrine for maintaining mission capabilities and readiness of soldiers. A minimum of 6 hours off duty during regular operation. Contingent operations I have experienced lasted at the longest longest 42 hours of continuous mission requirements. Contingency operations are entirely mission accomplishment orientated and sometimes the chain of command is unaware of the actual scope of the missions tasked. I call it clearing the desk; when you, as a leader, pass mission taskings off without analyzation and there for lacking strategic planning for sustainment.