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?

All I'm hearing is you have a problem with authority always looking for a way out and can't be trusted by your team mates if you get deployed to a hots spot like Kuwait. Go to PT!

Ummm.... I live my life on 5 hours of sleep...
What are you going to do when you’re up all night with a sick kid? Are you going to tell your boss you can’t get your job done? Are you kidding? Get with the program. Figure it out! You sound like a snowflake!

Hey this is Military Life, I get it but again this is military life. If you really want to brake it down you can sleep the 4 to 5 hours before PT then eat and go back to bed and that will give you at least 7 hours sleep. Your hours don't give you that night time play but you do have choices

Suck it up. Go take shower and go back to bed.

In Nam sometimes we would get only two hours of sleep and an alert would sound and then no more that night. Sometimes you sleep two and two off and then go fill sandbags, go on patrol, or whatever with no rest during the day. You could hump the boonies all night and then work or fight the next day. Bootcamp should have taught you that you can do more on less than you think you need. That means, less food, less water, dirty clothes, and no bath unless it rains. Accept and expect those facts or get a discharge before combat. After this week it may be very close and deadly. I think it was Tet of 68 when a nurse I grew up with was in Tokyo hospital and she told me later when we were both out that she went on duty one day and the wounded were flying in by chopper one after another after arriving in Japan that the rooms filled as well as the halls. The nurses were on 24-hour shifts. She was exhausted when a person on a table at night asked her to wipe his face, she almost said something nasty but stopped herself and saw he had no arms. She went into a room and cried. Her name was Bonnie Watson. Yes, she is real and she survived but was burned out by the pain, suffering, and stupidity of war. War is for the young stupid men who do not question to fight for the rich and powerful to profit from.

Suck it up or get out. Quit being a whiner.

It's PT, not a PT Test dude! Come on! Seriously, on occasion we'd come in drunk from the evening & straight to PT. I was 23 & didn't puke, some of the young ones did.
Considering your duty shift, if your NCOIC wanted to, the whole shift could do PT together right after duty. PT is a mutual obligation your branch shares with you to meet consistent fitness. Think it out, is it important enough to you & your shift persons to request? If do, approach the NCOIC through Chain of Command. It's a valid request, for a valid consideration, to a valid solution. If requested properly, it will be seriously considered.

I really liked my time in the Army, with one exception, lack of sleep. It wasn't just me, it was my company in general. We got pretty good at taking power naps when we could. When I became an NCO it could be even worse at times. I remember being up for 36 hours straight once during a field exercise. Learn to be creative if you can't work something out with command.

Suck it up and drive on. When you will get even less! Good training!

The fact that something can be done, or has been done a certain way in the past, is a poor reason to do it going forward. It is certainly a potentially relevant factor, but does not suffice as a sole criterion to justify a policy.

For better or for worse (and it seems there are a lot of those on this forum who would consider it for worse), the Army has recognized that maximizing human performance requires changing some of its practices. To this end, the Army as an institution and profession is promoting what it calls the "Performance Triad" which includes adequate sleep as one of three essential elements for maximizing performance.

Can we "suck it up" to get through a mission, operation, surge shift at the hospital, or whatever? Sure! But evidence-based research demonstrates that sleep deprivation, especially over prolonged periods, leads to sub-optimal decision-making and that leads to sub-optimal performance.

the leadership of a medical unit should know this better than any other kind of unit, as the Performance Triad is heavily promoted by the Army Medical Department. See, for instance, https://p3.amedd.army.mil/performance-learning-center/sleep, a resource that can help identify factors that lead to sleep deprivation and educate leaders on the possible consequences and steps they can take to mitigate and reduce the risk. This is something that not enough Soldiers or units take seriously.

As to the original question of how to bring up the issue. The Performance Triad is (or should be) being taught through various Professional Military Education courses. I also saw it in a pre-command course for First Sergeants and Company Commanders some years back, though I don't know if it's formally part of the program of instruction. Master Fitness Trainers should certainly be in a position to advise the leadership team on the topic, my understanding is that they are trained on the resources available through Army Wellness Centers. However you bring up the topic of the Performance Triad politely and professionally through your chain of command, there are institutional resources to help educate them on its importance and their role(s).

Separately, if you find that sleep deprivation is impacting your health (it can and often does), identify that to your own primary care manager independently or through your period health assessment. It may be that the time you have really is sufficient, but you have something else going on that results in your sleep not being restful enough. A sleep study would identify if there is a larger concern that needs to be addressed for you.