That panel needs cleaned up a bit. Wiring is a bit haphazard.
You only say that because you don't see all the neutrals runted off to barely make the neutral bar, and the hots runted off to barely make the breaker they're on now, with that cute little 90 degree bend. I follow a different rule.
Hot and neutral of every circuit can reach any space in the panel. That way you can move things around, add GFCI/AFCI when the inspector writes you up for forgetting all those, etc.
OP may regret following your advice.
you are installing a 'sub-panel' that is larger than the breaker panel it is pulling from, so no
You keep repeating this, and it's completely wrong. It is a huge misconception about how panels work and are specced.
And it drives people into the serious mistake of using far-too-small panels. The ones you linked
aren't even panels, they're basically fused disconnects that use breaker hardware.
I wouldn't even put one on an air conditioning unit because you need a third space for the serviceman's receptacle. That's where a 6-space would come in, that way when the guy comes back and says "I also want to power a mini-split here" you say "Gotcha covered" instead of saying "Well gosh, you'll need to replace that subpanel".
For the price difference ($10?) It's very cheap insurance.
Running out of space is a pain in the keister. I can't tell you how many questions I've fielded that start out with "My panel is full, and I need to add------" It's so stupidly unnecessary to have that problem.
And no, it's not about the $40 panel. It's about the hours you spend swapping it.
Since your house panel you are connecting to is only rated for 100amps, you can't install a downstream 'sub-panel' that is LARGER than that
Again that is the thing you are wrong about. What you say is simply not an issue.
The way your brain is working on this, if you bought HR rated tires, you'd feel *obliged* to drive 112 mph.
SUBPANELS DO NOT HAVE A SIZE. THEY HAVE A BUS RATING. You are misunderstanding what that number is. Look, a 225A panel does not draw 225A. Suppose you install a 20A and I install a 225A. We both put 13A loads on our panels. How much load does your panel
actually draw? 13A, not 20A. How much does mine
actually draw? 13A, not 225A.
The draw is decided by the load, not the panel.
Now suppose both our subpanels have 20A feed breakers in the main panel because we are power-limited like OP. What happens if you put 40A of load in your panel? The feed breaker trips. What happens if I put 40A of load in my panel? The feed breaker trips.
We both plug in a load that has a problem and pulls 150A. After 3 seconds, our feed breakers trip. My subpanel's "main breaker" doesn't even notice it. Your panel is on fire because of the cheap 20A busing.
Get it?
In a current-limited situation like OP's, the thing that matters is the feed breaker that supplies the current to the cable run to the outbuilding. The size of the wire does not matter. (as long as it >= feed breaker).
The size of the "main breaker" in the sub does not matter.
The bus rating of the sub does not matter. (as long as it >= feed breaker).
Keep it simple and only use this small one ... and for only a few circuit breakers, probably only a 15amp and/or 20amp for 'normal' usage for lights, outlets, power tools, etc.
Painting yourself into a corner is not simple at all. It creates a big mess in the future.
only needing MUCH smaller wires to the shed
Look. If you have a 20A feed breaker supplying a 225A subpanel, the wires only need to be #12.
Really. Because the feed breaker is 20A. You're always welcome to use larger wires than are required, so feel free to run #1Al feeder wire. Then when your service is upgraded you can just pop the #1 onto a 100A breaker. Really. It's that easy.
You make it much harder than it is. You keep thinking that's cheaper, and it's not.
the nice thing about this 'main lug' sub-panel is that it does not require it's own 'main 240v breaker' since it uses the main breaker you already have on the house's panel, for it's cut off.
No.
The disconnect switch has to be at the outbuilding. The cheapest way to do that is select a subpanel that has a main breaker, and you just use the main breaker as a disconnect switch. However since OP did not get that kind of panel, OP will need a separate disconnect switch.
Those 2-space "panels" you linked will suffice, just put a breaker in them >= the feed breaker. Or < the feed breaker. It really doesn't matter, the feed breaker in the main panel is responsible for overcurrent.
NEC 2020 now requires an external disconnect at services.
It does not require an external disconnect at outbuildings with subpanels.
You only need two individual circuit breakers within this panel, or you could use 4 'tandem' breakers, which would also fit on the two slots provided, giving you four circuits, if you need them.
2 years later: Help! I need to add a circuit and my panel is out of space. Also, if you have any 240V loads, your "tandem/quad" plan won't work. 240V loads will need GFCI protection, and that can *only* happen at the breaker.
And here's a thing you don't know. You're not allowed to run 2 same-voltage circuits to an outbuilding. Your logic is "when I need 2 more circuits I'll just run another 12/3 with another pico-panel". No you won't

You'll have to replace the kaboodle with a new #8 run, and have a proper panel.
For example, if you have two 20amp circuits within the new panel, you can easily pull 12/3 wire to it, since #12 is designed to easily handle 20 amps. The black wire to one lug, the red wire to the other lug, and the white to the Neutral lug, and the ground to the installed ground bar lug. Since the hot wires are each only serving ONE circuit, neither circuit will need anything larger.
Why on earth would you do that, when you could just run a MWBC. Then you wouldn't need a subpanel, outbuilding disconnect switch, nor ground rods!
Isn't this supposed to be about being economical?
You'll also be dead-ended, but you don't seem to mind that.