Powering Subpanel from CSED via Feed-Through Lugs
Conduit would be a good idea, even if not strictly required

The main advantage running wires in conduit gives you is that you can put fatter wires, or more wires for something this large even, in if you find that your original run was insufficient.  In fact, a 2.5" EMT provides plenty of room for 3 250kcmil aluminum XHHW-2 conductors for now, and even expansion room for 3 more of them should you, or the next bloke who wants this place, want a 400A service later on, in addition to being the obligatory equipment grounding conductor.  If EMT is too costly an option, 3" schedule 40 PVC run within the structure could work as well, although you'd then need to run a separate equipment grounding conductor.

If you do go with the cable route, I would go with a 250kcmil-250kcmil-250kcmil aluminum SE-R cable; this gives you the full 200A, but limits you to that all the same.

That's an awfully small subpanel you have there

A 30 space panel is rather on the small side; 42 or even 60 spaces would be a much better use of your money and time here, given that it's far cheaper to buy the spaces now than it is to rip out and replace a too-small panel down the road.  Also, you can use a main lug panel on the inside of the house as it's a subpanel in the same structure (so it doesn't need to provide its own disconnecting means) and is not fed by downsized tap conductors (which are protected from overcurrent by the downstream OCPD).