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?

It looks like you have plenty of time to sleep from 0830 - 1500.
I recommend you readjust your sleep schedule and sleep after PT, before work.
As others have pointed out, deployment is going to offer you a more stressful schedule and afford you less sleep. It's difficult to do, but you gotta drive on with a positive attitude. Try and look at it like a learning experience and see the glass as half full. That way when you do deploy you will be better adjusted to the "worse" conditions. You'll shine while others are lagging behind. You can pick up the torch and help others.
EDIT: I just read through some comments. Yes, it sucks that you have to schedule yourself differently than the first shift personnel. Correct it is not fair. Yes you'll do better on a more well rested day. But these difficult situations do not last forever. These difficult situations teach you a lot. You will learn from this experience and you can let it turn you into a 'negative person' or 'positive person'.
Life isn't always fair. Hard times end. Look for the silver lining.

myself i'd do it anyway !!!! that way if you pass you'll feel better about yourself !!
dont put off what you can do today

SPC(P) (Join to see) SUCK IT UP BUTTERCUP! That's nothing. Pack your PT gear the day before and have it with you. Work, get to sleep ASAP afterward, wake up, drink some coffee to get the motor running and do it. Going to your command over this will end badly.

I worked 5 hours Wednesday (my shift for a gate that closes at 8pm, it relieves the morning shift and assists the closer), 8 hrs on Thursday 1400-2200, 16 hours 0600-2200 on Friday, 16 hours 0600-2200 on Saturday and 1400-2200 on Sunday. Normally Friday and Saturday are 0600-1400, however, because of the likelihood, I will be off from Thanksgiving, I got to make up the hours. I have to take a dog to the vet at about 0730 tomorrow for a Cushing's test, look it up. Also, we don't get breaks or lunch. I maybe get 3-4 hours of sleep on Thursday/Friday if I am lucky. I am also studying for my BSIT Network Operations and Security degree at Western Governors and my final cert class before Capstone is CCNA Security with an absolute deadline of Feb 23 by Cisco.

https://www.wgu.edu/online-it-degrees/network-administration-information-technology-security-bachelors-program.html

I am F'N exhausted right now. Still, have 2 chapters to cover in 2 other books AND videos to watch just to get caught up before my degree mentor calls me tomorrow afternoon. I also have labs to do as well. Welcome to the real suck, it's called life. Worked 31 hours straight after working a 12-hour shift, off at 5 am, drove home 35 miles, washed up, dressed and worked from 0600-2030 as a Harris County (Home to Houston, Tx) Election Official doing field tech work with the electronic ballot boxes.
I had to make 5 laps covering all the sites in my district. By the time I got home and to sleep at 2230, I was up for 31 hours straight. The dropoff was in NW Houston and I live in SE Houston. Coffee and Red Bull were in my hands at any given point.

You are in the military. Suck it up cupcake.

And you are a promotable SPC (at the time you posted this) and whining (yes you are) about going to PT at 0530hrs when you get off shift at 0000hrs. When I was in, I been told (not sure if this is correct of not) that the Amy only has to give you 4hrs of rest, and those four hrs does not have to be all in one either. When I was in Korea (2003-2004), for 3 weeks straight, I got woke up after 2-3hrs to do a PT test. The commander wanted to get his numbers higher (I was not the only one and I was a SSG). Was I happy about it? No, but carried on and not bitch about it either. As they said in the movies, suck it up buttercup.

How many in your section is effected by this? If more then 3-4 people, why don't you take the initiative and come up on a plan that you conduct PT to your "squad" Brief that plan to your command (PLT SGT, 1SGT, CDR). The army wants you to be a leader (or you would not be promotable) and need to take the initiative, whether it is good or bad.

I know this been posted 6months ago and you might be a SGT by now. Hope it work out for you.

Adapt! Need to take your butt to bed early. Your not at home any more where you may stay out all night long and sleep till noon. This the military and only require to provide you with 3 hots, a cot and 4 hours of sleep.

Obey orders

Obey orders