Part 1: Know your loads
Unfortunately, I am currently renting, and the homeowner refuses to allow any work to be done.
I feel you. Same situation here, except my service is 120V/30A split to two 20A circuits for the whole house.
We trip a breaker several times a day.
STOP. DOING. THAT. Right now!
When a breaker trips, reset it ONCE. If it trips again, leave it be for an hour. Why? So the wires inside the walls can cool down.
We had the same problem with kitchen appliances, heater-fans to warm the bathroom and office, all stacked on top of one another. I sat my sweetie down and I said "Let's talk amps." I showed how to find the nameplate on a toaster or heater-fan, to look for "amps" in the 1-13 range or "watts" in the 10-1500 range (and divide by 120). The toaster is 8-ish amps, the heater-fans are 12.5 (unless on low, then 6). The last trip (and some intentional shut-the-breaker-off testing) told us which outlets and lights are on which circuits. Now it's straightforward. Crunch the numbers. Toaster + heaterfan = set heaterfan on low to allow some headroom (limiting pull to 14A).
We do the dance. Haven't had a trip in gosh, when was that, I wanna say 2016?
I have lots of experience in wiring things the right away, but I am not a licensed electrician. I really don't want to do this, but I don't really see much of a choice. I'm wondering if it can be done safely.
The main floor of my house only has two circuits to run four window air conditioners, 2 desktop computers, heat lamps for 2 reptiles, ceiling fans, microwave, air fryer, Keurig, fridge, etc.
Do the dance. If necessary put stickers on every appliance indicating its amperage draw (either derived from nameplate or from a Kill-a-Watt power meter). You'll quickly discover the reptile lamps, ceiling fans and fridge are basically rounding error, and you'll identify certain specific appliances which are piggies. (generally, things which make heat).
As for the PCs you'll need to get a Kill-a-Watt and measure actual load in different states of operation. Idling or word processing will be quite different from running CS:GO at 120FPS and 4K or whatever.
In the kitchen it's pretty straightforward. You just rolled down a list of heat appliances. The default draw is 1500W (12.5 amps), and the microwave may go as high as 1800W (15A). Because that's the UL hard limit for plug-in appliances. In our house the kitchen is super easy, only use one at a time. And in your case make sure the other stuff on that circuit is below 3A and you're golden.
Breakers have a fairly forgiving trip curve. They're designed to allow short term overloads since wires take time to overheat. So when you are tripping, you're way overlimit.
You don't have a prayer until you get conscious of your loads. Then it's just a numbers game. Make sure there's room on the circuit for what you're about to run. Walk the house, pause 2 air conditioners, do the kitchen stuff, turn em back on. I bet the PC is rounding error if you're not gaming.
For us, knowing our loads solved the problem.
I'm glad your onboard with avoiding hokey-dokey
Like this thing here you mentioned, there are 2 sources for those suicide cords. That one is from Cooter down in Fool's Hollow, who knocks them together on order. Cooter doesn't have any assets, so when that rig kills your wife, you have no recourse. The other source is China, good luck suing them. Of course, eBay, Amazon Marketplace, Banggood, DealExtreme, AliExpress and other direct mail sources are full of toxic dangerous junk. Avoid all of it.
The zoro.com item is an electronic component, and you can't use those directly as mains wiring equipment. It's either more cheap Cheese or it's RU Recognized, meaning if you were getting equipment UL-Listed, UL would ignore that component since it's preapproved.
Grounding is the biggest issue. That's what prevents you from just grabbing any random PC-grade PDU (that is UL-Listed).
One alternative to grounding is to fit a 2-pole GFCI breaker, change the socket to grounded (NEMA 14-50) and label it "GFCI Protected/No Equipment Ground". Then a 50A PDU would plug right in.
Part 2: the hard way
Honestly, if it were me, and I could not solve the problem with load management, here is what I would do. All of this is a big fat code violation - let's be clear on that.
As it happens I've been looking at very compact panels. Square D makes a "QO" 8-space barely bigger than a sheet of paper. I'd start with a piece of plywood and bolt that panel to it. I'll leave the layout to you. Then I'd come off it with 1/2" EMT conduit or just conduit nipples, out to 4-11/16" (120mm) square boxes, as many as 4 of them.* Power can pass through boxes, so feel free to put a 2x2 array of them on one side of the panel. Don't mount boxes flush to each other or you won't be able to get the cover plates on lol. Many nipples give you 1/4" or 1/2". Screw everything firmly into the plywood; use short screws.
I would get 2-gang mud rings* or Decora domed covers* for those. The mud rings can be pretty flat. These metal big deep boxes are specifically for the space two GFCI+receps take up. (you don't wanna know the price of a GFCI breaker in QO).
The GFCI receps will ground through their mounting ears. No need to wire a ground.
Through the conduit, I'd run black white brown gray. The electrical supply house will probably sell you the gray by-the-foot. Gray is the "alt" color for neutral. So black+white is circuit 1, brown+gray is circuit 2. If you need circuit 3+4, those are red+white-tagged-red, and blue+gray-tagged-blue.
This rig can support up to 8 circuits, so with 15A circuits that'd be (nameplate) 60A per leg which is overkill because we provision circuits on the assumption that only some are used at a time. Even eight 20A circuits (nameplate 80A per leg) is totally fine on a 240V/50A feed.
You don't need any grounds because the receps ground through their yokes and EMT carries the ground, but here's the important thing: Do Not, DO NOT connect neutral to ground ANYWHERE. The steel boxes are not GFCI protected, so we want the "grounding ha ha" in this system of metal totally isolated from the neutral coming from the NEMA 10-50. Because that neutral can be dangerous! That's why NEMA 10-50s are banned. That means remove the ground strap/screw from the neutral bar. If leakage happens between hot or neutral and the chassis of this thing, you're up the creek pretty much, because it's on the wrong side of the GFCIs to save you. So I would look for a way to independently ground this chassis.
To bring power into the thing, I'd use a standard NEMA 10-50 range cord and land the wires on the hot and neutral lugs. Take the range cord to the electrical supply and have them sell you an appropriate strain relief to enter a knockout.
That's about the best I can see to do. It's bad, it's Code-violating, but at least we're doing our best not to kill you.
* Get these at the electrical supply, they're twice as much at big-box.