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?

Can't hack it, get out. The mission may call for days without rest. maybe 10 mins of sleep is a delicious nap. Suck it up or get out of my military.

Suck it up buttercup! Apparently no one told you that you’re in the Army and we have no room for snowflakes! You are supposed to be a warrior..... at least fake it.

Somehow I missed this thread the first time around.
From a 1SG perspective, I can infer quite a bit from the 1SG's directive. There seems to be a culture in the unit where showing up for PT is optional. Five troops at PT is UNSAT. We don't know from the OP, but I'll wager another driver of this might be that the unit is not the best at APFT scores.
If my supposition is true, then the First Sergeant will make his point and then relax.
It is apparent that the 0530/1600 PT schedule is meant to accommodate two of the three shifts, and that it is illogical for some reason or another to conduct organized PT at midnight.
In the meantime, you're going to have to adapt your schedule, which shouldn't be all that hard with an 8-hour work shift. But I think that what makes sense is to have a pow wow within your shift to find a good course of action to propose to 1SG and let him know that his NCOs have got this and he can back off again - where any decent First Sergeant would prefer matters to be.
Since some time has passed, I am curious what the outcome was SPC(P) (Join to see)

Go as ordered snowflake. You're with the big boys now. NO WAY OUT. Following orders is expected. Did you learn nothing in basic?

From a Marine Vet that was 0331 M-60 gunner I have to SMH. Really? If you cannot PT on 5 hours of sleep then what shape are you in if you have to deploy? The lives of your brothers and Sister could depend on that, the mission could depend on that. I have to ask if the military is suited for you?

I would say your lucky to have gotten so much sleep!! If I, as a 1SG, had only 5 show for PT I'd be a little ticked off also. Suck it up for now, sometimes the squeaking wheel doesn't get oiled. Put up with what your given regardless of how unfair it may seem. Look at it like a test, and your being watched. Come time for a promotion don't be surprised if you find your name on that list!!

Life's a bitch and then you join one.... then you retire and get to sleep whenever you want

SPC, as a guy who served actively for nearly 40 years, officer, NCO & enlisted, I will tell you that you'd better get prepared for more of these days. When you go into operations, your battle rhythm doesn't always allow for "oh captain, I'm not going to get my 8 hrs of sleep, you need to change it up." Bud, in the heat of uptempo ops, you may have a long string of nights where you MIGHT get 4 hrs. So, my advice to you is work on ways you can adapt to operating with less sleep. It may come in handy one of these days.

Unfortunately, the non-combat world revolves around the day shift folks... good or bad that’s just the way it is. As a hospital administrator I understand your desire to sleep in after a swing shift. As a military member though, we’re sometimes called upon to make some sacrifices that on the surface don’t necessarily seem fair. Keep in mind there’s a bigger picture playing out that’s bigger than any particular person’s needs or wants. Military must be fit... bottom line. Our missions require us to be at our best, both physically and mentally. Suggestion... as others have said, suck it up for now. It won’t be forever. We all have our time in the barrel, but 99.9% of us make it, and we’re better men and women in the long run for having made it successfully through a tough time. Keep in mind... at least no one’s shooting at your ass as you run... there’s worse things than being sleepy.

Just sounds like you should deal with it