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?

I'm guessing that you're not an Airborne Ranger....

Suck it up troop, this is not camp. Definitely sounds like quibbling, if not whining. Move out and drive on.

You should handle it by doing what your told it’s a MWF schedule... suck it up buttercup there are thousands doing your and doing it well with less sleep than that!

You have two options. 1st is to suck it up and do the first slot. 2nd is to negotiate an alternate time, but make sure you keep your word. If 2nd doesn't work, don't complain and just do it.

If you feel you're not getting adequate recovery time and it's affecting your physical well-being, bring it up to someone. The worst that can happen is that nothing changes. If your fatigue becomes a safety issue, then you're a liability. There isn't much in the way of specificity for RA soldiers in garrison though.

FM 22-51 Appendix A-9 states that while 6-8 hours of continuous sleep is preferred, They're only required to allow for a minimum of 4.

It is also stated in FM 6-22.5 Chapter 4:
Continuous operations are military operations with many pulses
of action every day and night, continuing for several days to weeks,
which require careful planning and resource allocation to give
everyone a minimum of 4 hours sleep in 24. (FM 22-51)

While this doesn't specify garrison, I wish you luck in getting that message across. It may be an easier task in your world than it is in mine.

TRADOC Regulation 350-6

(1) In garrison, IET Soldiers must have the opportunity for 7 hours of continuous sleep per night, unless the Soldier is scheduled for duty (for example, access control guard, or CQ runner). During field training, the length of the training day and time for sleep will vary based on tactical and other training requirements, as established in the appropriate POI. Commanders will ensure that the tactical framework of field training allows sufficient time for rest.

(2) In garrison, Soldiers are given at least 1 hour of preparation time each day to take care of personal needs.

Hope this helps with something.

There are 16 hours between your shifts and you can only find time for 4-5 hours of sleep? What will you do if you’re assigned to a combat hospital on deployment? You may get less than that.
What I see is less of a PT problem and more of a you not wanting to adjust your sleep schedule problem. You want your free time to be during the daylight hours where you can socialize and have fun instead of having free time from 0001-0500. The problem is that the military does not conform to your schedule, you find a way to work around the military’s schedule. If you cannot figure this out, you do not deserve to be promote past your current rank.
There will be many times in your military career where sleep is a rare thing - why do you think soldiers can fall asleep anywhere? (even in the back of a deuce and a half) Your job is to learn to adapt and overcome.

Also, best way to learn things is to look them up. Instead of asking about Army Regulations - look them up yourself. You will be amazed at how much information you retain.

Working at strange hours and also conducting PT is a normal part of military life. Achmed, Ivan, and Charlie won't excuse you from combat because you lost some sleep, and your physical conditioning is as important as every other soldiers'. Your section, presumably has an NCO. Top and the CO expect their troops to PT but I doubt either of them really care when that takes place as long as it's quality training and it actually happens. Perhaps your NCO can ask Top to allow your section to conduct section PT at 1330. That would leave everyone time to shower and get to work by 1600.

Poor baby. Here's your binky--Mommy will bring you some hot cocoa.

From a Commander and First Sergeant perspective I understand the everyone comment and have everyone show up. However, that is a measurement with a micrometer marked with a crayon and cut with a chainsaw. The CC and the 1Sgt are right BUT they probably don't know the entire story. You probably need to work it through your platoon leader and up and see what happens.

NOW WITH THAT BEING SAID...you work 1600-0000. You get up for the 0530 PT formation for lets say an hour and that takes you to 0630 and you go back home. That leaves you 9.5 hours to rest up for your next shift AFTER the PT. I really don't see an issue here. There is plenty of time for you to rest (couple of hours prior to PT and 9+ hours after.)

You get no slack working in the hospital! Take it from a USAF nurse! I then went onto be an AFOSI Special Agent, Forensic Consultant and Defense Attache and got even less sleep typically 4-6 hrs. During 9/11, I was in Bosnia doing peacekeeping and went to 4 hrs of sleep for 3 months taking down Al Qaeda terrorists with Special Forces and supervising US Arabic translators translating seized materials and preparing briefings for the President. If you can survive on 4-6 hrs, it gets a lot worse as you rise in rank!!