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?

Great answer CPT Parr. To easy for supervisors to relay that information to the Platoon Sergeant so that he/ she may talk with 1SG and commander for an exception. To easy to have an NCO on that shift conduct PT prior to the 1600 work call. And for accountability have a sign in roster that is turned into the 1SG on a daily basis. For those in the post that have never worked in a hospital you don’t understand the intricacies of different shifts. I have been in hospitals that have tried to do mandatory PT and I can tell you it didn’t last long. What is gonna happen is higher level command will get involved, and/or you will get and IG complaint. It is to simple to let NCO’s handle this at the lowest level. All NCO’s should know how to conduct PT. Come on everybody don’t be so narrow minded to easy to improvise and still complete the mission. One other thing. Most medical personnel in hospitals are conducting a real time mission everyday. As a retired NCO whom spent time in TDA units and hospitals I can tell you it’s two different worlds. I have deployed with a CSH and a coward surgical team and how your work is stateside and vs. deployed are not indicators of how you carry out your the mission when deployed. When you are talking about saving lives you be surprised how seriously our service members take it. For those of you that have deployed you know what I am saying.

People like that is why the future of our country is fucked

Well said.

I don't recall hearing that in the entire history of the world anyone ever dying from lack of sleep. But, on the other hand, many have died from being unprepared. I think your NCO's are just trying to make sure you don't die for your country, but that you'll be able to make the other SOB die for his.

Sorry if that sounds cruel but I am Airborne after all.

Why needlessly suck it up when there are, or can be options? Telling people to "suck it up" for an entire enlistment because it's too hard to be good leaders is why those same people get out of dodge when their term is up.
It used to be that we took care of our own. Now we just say "back in my day..."

You're comparing "those times when the mission demanded little to no sleep and I had to suck it up" to "for some reason the command made work shift for some people with PT to be scheduled 5 1/2 hours after that shift ends. Every single day." I'm pretty sure if you worked your standard shift and were then called in 5 1/2 hours later every night, you'd question the need for that scheduling.

CPT Michael E. Cain - "Back in the day?" And when did you mobilize for DS/DS? And what was the huge machinery in your unit? The biggest part of the HAWK missile system that I remember was the radar unit.
And 'back in the day' in most units with 'good leadership' the soldiers described with this issue would have been given a third option for PT, with the 1SG periodically checking.

SSG Robert Webster - I was in the forming HAWK Battalion (2-263 ADA) in the SCArNG during DS/DS. If my memory is correct, we were on a 4 hour call up. Again, from memory, I believe we were originally tasked with personnel replacements for the 18th Airborne Corps HAWK assets, for when the expected casualties started. Luckily, that was the closest I ever got to combat.
That being said, I served in a Duster unit in Virginia, (D-3-111 ADA), and the 2-120 Mechanized Infantry, attached to the 4ID. I promise you, we trained at night, around large moving equipment with little to no working NV, and when sleep deprived.
In the "real world", several of the senior NCO's in this discussion have already proposed a decent solution. Those on back shift can have an alternate PT schedule. Everyone does PT, and it doesn't interfere with their sleep schedules.
Did you and I work together at some point?

CPT Michael E. Cain - No, we did not work or serve together at some point.
Number 2 - You need to correct your profile on at least one point - Iraq did not invade Kuwait until 2 August 1990.
Now as far as your comment about sleep deprivation around heavy equipment that you were familiar with. Well I know that during my Basic Training in 1976 at Fort Knox, then home of the Armor School that 'ground guide training' was mandatory for safety reasons at night; and then when I went to Infantry AIT at Fort Benning (again in 1976) that 'ground guide training' was also mandatory during the M113 training, also for safety reasons. And when I arrived at Fort Bragg and the 82nd Airborne Division, when we had exercises with the 4/68th Armor (or 3/73 Armor/Cav) with M551s it was mandatory to have ground guides when operating around dismounted troops at night. So I find it hard to understand your point in that regard. I do know that back during that period that when people failed to follow those safety guidelines that there were an inordinate number of preventable injuries and deadly incidents.
The problem illustrated in this situation is that what should be an acceptable 'alternate' PT schedule is not available.
And as far as a number of the suggestions given so far, most do not even consider the ramifications of a bad lab result due to sleep deprivation and how that would impact a patient, the soldier, and the soldier's chain of supervision and command. With that in mind, most of the answers given are total BS.