I've been staring at this for 10 minutes, and the more I see, the more I marvel at how this contractor not only nailed the job, but nailed the customer. Every detail aligns not only with best practices, but your peculiar requirements, and level of knowledge.
Rest assured this contractor heard you 5x5... the issue is, you didn't hear the contractor, probably because of the snow of technical information inherently involved, and, all due respect you are one who is hasty to judge, as seen in your question...
And again all due respect, the contractor surely tried to explain things to you... but that wasn't working, so the contractor did exactly what you said, because you are the customer.
Provisioning 300A to the house
With 400 amp service to the home, they put 2, 200 amp panels in the home. One is about half full, the other about one third full. Again, no idea why they did this. Only now an I realizing what happened.
Well, oversubscribing a panel is really no big deal, unless you have specific loads of known ampacity that would overload it. That's because in a residential context, nobody uses every circuit at max all at once. Most circuits hardly see any load at all.
But you have the Leviton board, and I gather you'll soon install all the smart innards... so you'll soon see for yourself how farcically less than 400A your house actually draws in practice.
If you hang around here awhile, you'll see us drum to death the idea that you should get BIG panels, and the bus rating of panels can, and should exceed the breaker trip supplying it, both so you have plenty of spaces because running out of spaces is expensive... and so you aren't running the buses at redline. If you drove 85 mph, would you want 85 mph tires?? Of course not.
As you know, single 300-400A panels with 80 spaces are cost prohibitive, so it is standard operating procedure to use two 200A panels. So the fact that you are seeing 2 panels instead of 1 panel is no concern. But this installation used a meter-main with separate breakering of each of those panels. You know how you insisted on a 300A/100A split? Go look at the 2 feed breakers on the meter-main. They are 150A each. This is exactly what you asked for: 300A to the main panels and 100A to the shop.
I bet the panel didn't come with 150A breakers. I bet they had to order them in special. I wouldn't have batted an eye if they were 200A because oversubscription is okay, in fact ThreePhaseEel assumed they were, and the Eel is way smarter than me.
Where can the shop breaker go?
There's one thing about this installation that rankles me: the contractor wrote "spare" in pen. The whole point of a spare breaker is to have it do something else later! But this begs a bigger question: why install those 2 spare breakers at all? It's gratuitous and unnecessary.
That meter-main is really unique, in that it provides both a) dual large main breakers for feeding distant panels (as done here), but also b) 8 regular breaker spaces for this-and-that.
Those 2 "spare" breakers make it easy to notice: "oh wow, breakers CAN go here". Which I suspect is the real reason for the breakers. The contractor wasted $10 to light our way. Honestly I might have missed it myself.
Anyway, suffice it to say, your 100A shop subpanel can be fed straight off the meter-main.
There was no need for a meter-main of this sophistication, except to feed the shop from here.
All you need to do is come out of the bottom of the meter-main with some conduit and head over to the shop. Easy peasy.