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?

if you aren't working at PT time, get your rear out of bed. if you weren't working at midnight, would you be at the club until 2?

You are in the military and must follow orders, suck it up and due as you are told. Semper Fi

I feel the point is being missed here. SPC Davis has a high-demand job that isn't in a war zone. I believe she would step up her workload and go a lot longer without sleep if required.
But working every day with this schedule can wear you down. I worked for years on the 0000 to 0530 shift and that is a bear. Hard to go straight to bed without a little downtime. And why should she be expected to give up all of her time to PT and sleep?
I would take this matter up with my platoon leader first, then go higher if needed. Just be honest about how it is affecting you. If all else fails there is sick-call.

E-4 mafia solution go to PT and "sprain your ankle" doing jump-in-jacks.

You are complaining about your problem. If you read the regulations. It does not say what time you have to do pt. If your 1st Sgt. Says you will be at formation after your work your night shift. Then you will be at pt formation granted that is the end of your day. If you look at the time schedule of a typical military day 0500 wake up 0530 0630 pt breakfast in a hurry with personal hygiene 0730 to 0800 morning formation then an end formation around 1700.
Now flip it to an 8 hour night shift... you get off work at midnight then get 4 to5 hours sleep go to morning formation and pt then you are on down time until your night shift...
Where's the complaint?

For now, you go to PT. If you don’t, you will be guilty of failure to comply with a lawful order. And that can result in non-judicial punishment and a blot on your record. And if it continues, you will be subjecting yourself to more serious consequences. Go to your leadership at the hospital. You will fare better if the leadership addresses the concerns to your 1SG. As a junior enlisted, I hate to tell you this, but if you complain, you will just come across as “sniveling”. A person in your chain of command at the hospital will have a better chance of making your case.

I can remember days in AIT that I had to do midnight KP and fire guard on the same night, then be in formation at 06:00 for PT. Had to take a catnap during lunch, but I made it through.

Having 2 times to get to PT? What a luxury! I got around 3 hours of sleep before our PT. Only one a day at 0500.
How do I bring this issue up and solve this effectively and professionally? You don't. You do not miss formation or PT unless you are either on duty or out for medical reasons. Same thing for being late for, or missing, work.
Do you know of any Army Regulations that can support anything? Failing to show up for PT is a violation of Article 86, Failure to Report.

Get the hell Out!!!

Can appreciate it. Military has always had a problem matching career patterns with what a soldier should be doing and when. There is a difference between what the soldier should be doing and what he should be doing to get ready for combat. Both need to be rote when you get into combat. However that is a different stress scenario. It is a way of weeding out those you can adapt to a combat 24 hour and a 24 hour support situation. It is something which requires CO's at different levels need to stay on top of. time management is needed to the max. However administering & managing is required too. I dealt with supporting combat troops in RVN and it often took 18-23 hours of sleep a night. Keep in mind that combat is a 24 hour stress situation if you want to stay alive. Being support means the same thing, except if you want to not be combat while doing your job it may wind being over run. So you have to avoid becoming the front line and the support. Survival means a lot and takes a lot. It also means not having Officers & NCO's who keep a unit function to maintain that end.