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?

Go through your chain of command to bring up the issues. Everyone knows that while on active duty you have to do PT. You are lucky that you have a choice of what time you want to pt. It sounds like your 1SG is trying to make a point. If you have shift to work from 1600-0000 and have the rest of the day for yourself then do the 0530 and by 7 ish you should be done and in the rack.

I'm 74, at your age, I could get 4 hours of sleep, rise at 0400 and do hours of exercise, work a full day, party a little and do it over again. If you are under 60, suck it up, cause I still do it daily on 6 hours of sleep or less, running a horse rescue. I am sorry, you need to know you will never grow old if you keep your mind and body in shape, and get a better attitude. Your 1SG & CO are only looking after the health and welfare of their people. And for heaven's sake quit whining.

Your enemy is laughing at you.

Well, how about 1500 PT? I propose that, and tell the shirt you will maintain the sign-in sheet.

If your section sergeant feels the same way as you, then he/she should suggest to your 1SG/CDR if your sections could conduct PT at 1430 -1530, right before work, and ask them to join you guys in PT. The command is only looking out for your health, wellness and wellbeing. They understand that they inherited a company that does section PT because of the shift workers and l'm sure that your leadership attends morning/leaders mtg, they should be addressing the issue. Anyway, passing the PT test keeps you in the military and the only way to do that is to stay fit and to stay fit is to do PT. So it's part "time management" on your side and part "personnel management" on the command side
I use to work in the hospital, LRMC, so I've been there and done that with working clinic hours and shift work hours ER/ EMS.

Go explain it to whoever your supposed to report to for your shift, or the tippy top person that they fall under. Then ask them to go clear it up. If it's an officer that you fall under then your Top's balls will moat likely get grilled.... win/win

Ahhh…this new Army. I’m sure there’s a career out there for someone of your caliber who needs 8 hours of sleep. When I was in Iraq I needed at least 8 hours of sleep before I could do my duties. I just don’t understand why your command won’t grant your wish!

First, follow orders. Learning to do so will save your life one day. Second, assess the situation before you take it to the chain. Perform the function as instructed for at least 1 month. THEN provide feedback to the command as to the feasibility and or fatigue state of the crew impact. Be able to identify the negative readiness impact on your section. If you can't do this it will be considered whining or complaining. Your feedback needs to be for the betterment of the Armed Forces not for yourself. When you take it to the chain start at the bottom! Follow the appropriate chain of command - ask to have the issue elevated if it feels stalled. You will gain a better understanding of each member's issues and concerns along the way not to mention respect for doing so.

Just speaking generally here. In the USCG at least, only the mid-watch, (0000-0400 or 0000-0800 depending on specific unit or watch,) was exempted from morning quarters. 0000-0400 was allowed rack time until 1000, 0000-0800 was generally off until their next duty. On SAR stations, if you had a call that either began after 0000 or ended after 0000, you were generally allowed to sleep in as needed. Calls 6 to 8 hours or more on a cold, dark, stormy night are nothing to be sneered at.

Short answer, life is not going to bend over backwards for you, in the military or out, so embrace the suck.
You've got options here. Senior leadershipay not see it, but the fact is military life doesn't always run on a 24 hour clock. Sometimes it runs on 12 or 15. It takes a couple weeks to adjust to that.
On my deployments, General Quarters and main space fire drills took place in the dead middle of the off periods for the 2330-0530/1130-1730 watches and lasted a good 2 hours.
First option: run a 12 hour day. Up at 1530 on one, work til 0001, in bed at 0100. Up at 0500, pt 2 hours or whatever, then a mix of sleep and personal time til 1530.
The new command is throwing it's weight around, that's all. This crap lasts a week at most, then stops.
Second choice: PT on your own and don't show up at formation. This will probably get you an article 15. More on this in a minute.
Third choice: talk to your immediate supervisor. He or she should be able to find out what the new chain of command expects in PT formation, and then have your shift do their own pt either right before or right after work.
The best solution, though, is the first one, at least til your NCO gets a new formation time approved. There are going to be times when you're running in empty and more needed of you, day after day. In this case, the only consequence if you fail is an Article 15 for missing PT (don't scrimp on patient care to make it to formation). In a war zone, the consequences could include loss of your entire unit. If your chain of command never pushed you like this, you wouldn't be prepared for war.
Upper leadership shouldn't act all shocked that this happens. If your soldiers can't handle this, they're not fit for combat. It should motivate them to PT on their own after a few days, then the stupid PT muster time can be changed or dropped. If they don't respond to it, there's a process for separating those who can't meet the physical fitness requirements. They don't deploy, and after the third failed PRT, they get an admin sep.
Civilian life doesn't always run on a 24 hour clock either. You might separate and find yourself working two jobs to make ends meet and have those exact same hours. In that case, they're not going to change just for you, one or both of them is gonna dismiss you. Civilian hospital employees are on call 24/7 and sometimes go a day or two without personal/family time and short on sleep. They're not on base with the hospital right there, they may have to commute 1 1/2 hours each way or stay overnight in an empty room.