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?

Old School mentality here. Suck it up butter cup. Several of the things I learned in the military was:
1) If you slid out from doing something, even if it’s regulation, it pisses off the person commanding it. If they are a vindictive prick, they will stick it to you a different way via regulation.

2) Tis better to suffer in silence and be rewarded in another area, then to bitch about something, labeled a trouble maker and assigned to the most Shitty duties that come down the pike. I noticed guys that just did whatever, regardless of how unfair or unjust received favorable treatment later on where the shitbirds just got dogged

3) Set the example, do the task. The military isn’t an easy or fair life. It builds character. You may need the example to help you out later on in your career. The military is ALWAYS training for war. Look at the guys that were in combat. Fatigue was a constant. Our Nam vets in the heat of the jungle, sweating there nuts off, humping up and down valleys on endless patrols, now it’s guard duty time, already tired and worn out. Watching your buddies getting some z’s. Shit! Got contact! Contact doesn’t break till dawn. Police up your brass and gear. Don’t leave nothing behind for the enemy. Now it’s time to start humping again. I’m not a Nam vet. I was trained by Nam vets. The more you sweat in training the less you bleed in war... Go to PT, go tired, do your job. You’ll have to dig deep inside you. You’d be surprised at what you can do if your mind is with it. A navy seal said, “ when you feel tired and can’t go anymore, your body is only 40% tired. It’s your mind telling you to quit because you feel uncomfortable”

I say you need to suck it up.

Certainly, the next war we fight will feature civilized enemies who will take into consideration such humanitarian things as whether or not we have had a good night's sleep, a nice wholesome breakfast, the weather to ensure no rainy or snowy days, and of course a recent mani-pedi. Physical fitness is in your best interest, and so is obedience to lawful orders and regulations. Perhaps your best recourse is to find greener pastures.

Unfortunately, like everyone else in your situation your day just became your night. You now need to get your sleep during the day before work. Good luck

Well, if you ever find yourself in a combat situation, you might be running and jumping, just like PT, in order to carry out your job. This might be done on no sleep. Just deal with it. You can do it.

Whatcha gonna do when you're down range and you're needed on duty 10 or 12 hours...how about 36? Stop crying princess. Learn to manage your time.

"However, my section is the only one in the hospital that has a 1600-0000 shift."
"everyone must show up to PT at either the 0530 formation or the 1600 formation. "

Looks like you are working from 4pm to midnight. That means you should be able to hit the rack until 0430/0500, after all, nothings going on after midnight, make the 0530 formation, shower, go back to bed until say 1000, get up and go about your day until 1530 to head for your shift.

I don't see an issue here.

I hate telling stories beginning with "Back in my day,..." but here it goes. In my time in 2ID, we'd generally have passes off-post until midnight. We'd drink, sometimes a lot. We'd often continue a party until 1a.m. in the barracks until the CQ would tell us to shut up. My aid station would have to be up and running by 0445 for sick call, then we'd PT at about 0600, sometimes still somewhat inebriated. So quite a few of us medics and other soldiers in the battalion would be doing PT on 3-4 hours sleep and possibly inebriated or hungover for or five days a week.
The difference between your situation and ours in 87 is that ours was self-induced. Same effect/result, but we didn't complain.
I also worked 12on/12off in a hospital sometimes, when we weren't on the three shift rotation and participated in unit PT, right after my duty day ended, at 0730 or just before it started depending on my work schedule.
You can adapt your daily schedule to accommodate PT, nowhere is it written that all your sleep must be in 8 uninterrupted hours

Looking at this objectively, I sympathize with your situation but understand the fundamental requirement of PT. Commanders intent seems to be the key piece of the puzzle here. What is that intent? Why are we emphasizing these PT formations?

That knowledge is paramount to your case as far as gaining an understanding of all parties goes. If these PT formations are being used for accountability, then wrong answer. There are better ways. If it’s being used for PT because “traditional army hooah we have to start every day with a five mile run” then also wrong answer. Mission takes precedence and as a medic your patients and your reliability and efficiently are insurmountably important.

Now if this is all because there are APFT failures or h/w failures, I can begin to see the commanders intent, but it could stand for some flexibility. Is this the only action available to achieve the desired result? What about encouraging small squad level gym times, and let lower leadership operate within the scope of the commanders intent? That’s what you have leaders for right? Your shift gets done, your head of shift dictates a training schedule that works for you and your shift mates, adhere to schedule, achieve results.

The only flaw I see in this is if it’s not like 92Gs where the same people are kept together for a “shift team” and the teams are rotated out on a daily schedule. If so I apologize for my lack of medical operations knowledge.

You guys have it easy lol. We only had to have 4 hrs of sleep that could get interrupted every 20 minutes. Not to mention when your overseas you get women up by explosions and gun fire. The military is supposed to get you ready for battle in a war zone. Your supposed to be able to function on little too no sleep. I've been out now for almost 10 years and I now drive truck. I still operate better on 5 hours of sleep. Put on your man panties and suck it up. I'm glad I got out when I did because of this very attitude. Your going to be one that gets someone killed in war because you didn't get your beauty sleep.