So, this takes place in a post apocalyptic world, within the borders of a government called the USC. 30 miles north of The Junction (modern-day Bakersfield) is the JSCF, or Junction State Correctional Facility.

About 2500 prisoners are held within the JSCF. The JSCF is used as a punitive labor prison, where convicts are used to do public works projects, such as building roads and railways.

The USC wants to use the smallest amount of guards they can, to cut costs. So, my question is how could a labor prison be run in a way that minimizes the amount of guards needed to run it?

The current technological level is near that of 1890s America, and the USC has considerable infrastructure

Basic layout of JSCF (please tell me if it isn’t realistic)

I just wanted to remind everyone, USC aren’t monsters. No abusing prisoners human rights

https://www.worldanvil.com/w/after-doomsday-dattawan

Turn prisoners into managers

Many societies across time have used this tactic (Romans for one, Nazis and Stalinists for more modern/malicious examples). Find the prisoners who are willing to enforce rules on other prisoners in exchange for more rights/luxuries/privileges. This creates a type of managerial oversight among the prisoners themselves. It's better to manage than to labor.

Extreme rewards and punishments

Any small violation of rules is enforced brutally. Beatings, tortures, etc. Snitches are rewarded handsomely with accommodations and protection from guards. This keeps prisoners looking for relief an easy way out of their rough situation: betray someone acting out. This type of regulation also keeps prisoners from trying to group together or make a consolidated effort at rebellion.

Group rewards and punishments

If a road or building is constructed well, everyone is rewarded with a day off. If one person makes a run for it, everyone in the group has to work through the night. This reinforces the idea of group obedience.

Keep prisoners occupied

Wake up, work, sleep. Repeat. Give them neither time nor energy to think about anything except their labor. Just enough food to get the work done, and only at the end of the day when they finish the labor. No lunch breaks. Keeping prisoners hungry and desperate for "one more meal" will deplete resolve and eliminate any practical hope of escape.

Although this also breeds intense resentment, there should be a clear path to eliminate or minimize their suffering. Snitching on others, pursuing management over other guards, and performing good labor are simple, safe, straight-forward paths to make their situation better. If there is no (reasonable) hope of escape, but clear paths "up" by turning on other prisoners, then prisoners will pursue those options. Make extreme examples of those who don't.

All these together allow for a few, well-organized, intensely brutal and malicious guards to control a large number of prisoners.

Inmates are the guards. This is the Trusty system.

https://en.wikipedia.org/wiki/Trusty_system_(prison)

The trusty system had designated inmates used by staff to control and
administer physical punishment to other inmates according to a strict,
prison-determined, inmate hierarchy of power...

The bulk of guarding and disciplining of the inmates was performed by
inmate trusties. They also performed most of the administrative work,
supervised by a few employees... Essentially, the trusties ran the
prison system. Highest in the prison inmate hierarchy were the inmates
armed with rifles, called the "trusty shooters." Their job was to act
as prison guards and control other inmates on a day-to-day basis in
the residential camps or out on the field work crews. Next came the
unarmed trusties who performed janitorial, clerical, and other menial
tasks for the prison's staff. Simple tasks, such as distributing
medication, were carried out by other categories of inmates such as
"hallboys.' Inmate trusties enforced discipline within the prison
inmate living quarters (16 different residential camps) and in the
work camps and farms.

Prisons using this system did it for exactly your reasons: limit required paid employees. Trustys were chosen as people who would recognize when they had a good thing going and not screw it up. Oversight was not a priority. American prisons in the South had to quit doing this this way in the 1970s but this system is still used in other countries.

Other matters, such as the importance of keeping the prisoners busy, have already been mentioned, but we've overlooked a historical consideration from 100years before your requirement.

The Panopticon

Designed by the late great Jeremy Bentham of being stuffed and sitting in the halls of UCL fame. The principle of the Panopticon is that a single guard can see into every cell in the prison from his vantage point. The idea was simple, to maximise the effectiveness of a single guard in a prison situation. Now of course we'd do that with CCTV, but that's surprisingly ineffective. Design your buildings such that there are no dark hidden corners and everything is visible from a small number of central locations.

One Guard

All you need is a warden to make the decisions and an automated system. Prisoners have obedience collars. Door are remotely operated. Cameras are everywhere and everything is recorded.

If a prisoner plays up the collar can do everything from an electric shock to blowing their head off.

Should a collar be removed, other prisoners can be employed to capture/execute the escapee through the threat of their own collar.

You already have a funnel. To me your biggest flaw is that your guard station is within the prison itself. If you positioned 2 guards (one necessary and the other as backup) at the funnel behind a door, then in theory you wouldn't need more. But your drawing doesn't make mention of what the clinic is, or really any of the rectangular identifiers.

Your governing process would be more important than the number of guards. You could have 40 guards and run the place badly and it would fail.

If rearranging the prison is possible - you did not say the prison couldn't be expanded or redesigned - you could move the clinic and guard station behind the funnel door, in a transient hallway, leading to an exit funnel that has no way to cross a hellishly deep and wide moat except a draw bridge from the other side.

If the clinics and guard stations had doors to both the main area and the hall, they act like a butterfly catch that prevents them from going through by way of one door not being allowed to be opened until the other is closed. Not beyond 1800 american technology.

Your kitchen is cut off except the service windows in the mess hall within the main room. Kitchen staff cannot interact with guards based on how you designed the food slot system. Use your imagination there, but I was thinking like a McDonald's burger queue. Slide it down a 20 foot pipe and your cooks never see the prisoners. No point of escape there.

All dorms are open so visual check is possible from the guard station across the way. Guards never need to interact with the main room. not even to break up a riot. Your governing system keeps them in check by threatening to lose food privileges or whatever else you want to leverage to keep them relatively sane.

So you have one guard in the guard station at all times. Another guard to accompany a doctor or whoever may need them without the station guard being away from post.

The draw bridge is only lowered by the convoy system that is used both for switching shifts or bringing in new prisoners, which go through the incoming butterfly trap so no prisoners can escape while new ones come in.

If your process was strict, and made sure of a couple things - at least one guard in the guard station at all times, only one door is ever allowed to be open at a time, never enter the main room, etc - you could conceivably run the prison with only 2 guards per shift. 3 shifts minimum. That would not include whatever convoy you have in place for shift management, which would presumably be in control of the elaborate draw bridge handling. But that's really just in the details.

Then you have to consider things like emergency cases where the guards need to flee, evacuation, communications, etc. But that's for another question.

If you can build deep within a mountain somewhere with only one way in or out, then guarding the facility would be easy. This would imply you have zero regard to what the prisoners do inside. If 2500 go to sleep and only 1000 wake up, then so be it.

In this scenario, the prisoners would be responsible for their own care. food and resources are delivered remotely, maybe via rope through a vertical shaft. The prisoners would have to distribute to goods amongst themselves. Now the goods are delivered proportional to the labor the prison provided. So if a prison gang wanted more goods, they would need to organize working situations for the others to maximize their profits. What does the guard care, if the gangs kill everyone else, or they starve to death, they get more "prisoners" to replace them until they start producing to requested labor.

The work they do in the community would be were the large number of guards come in. Maybe, to whom ever the prisoners are working for take responsibility of the security of the prisoners, it would reduce the labor cost of the prison itself. For example, corporation A rents the labor of 25 convicts. They provide their own guards. They receive 25 convicts, pay for 8 hours or labor and returns 25 convicts, or corpses.