Is feeding a 200A sub from a 200A main legit?
Yes, and you don't even need another 200A breaker. The 200A breaker alread present in the panel will suffice, and so you can use a subfeed lug kit as DrSparks advises.
For that matter, if the main breaker were out at the meter (e.g. a meter-main), you could simply "tee" off it with dual 4/0 to two main-lug panels. Panels don't need main breakers, though if they're in an outbuilding they need a disconnect switch, and choosing a main breaker panel is a cheap way to get a disconnect.
Is 4/0 correct?
Yes, 4/0 Al is sufficient. The reason for this is 310.15(B)(7)(1), which allows 4/0 for 200A service wiring due to a favorable derate. And then, 310.15(B)(7)(3), which says feeder never needs to be bigger than the service wires themselves, because that would be stupid. NEC is not about stupid.
Since your service wires are allowed 4/0, feeder off that service need not be bigger than that, even if it supplies less than the entire dwelling.
Is this crazy?
No, but it's unnecessary. You don't need to bring over the full 200A. The fact is, in any random home, if you subtract the large 240V loads - range, dryer, water heater, heat pump, hot tub and EV charger -- everything else will fit very comfortably on a 65A panel. If that makes no sense, it's because of the common belief that circuits each draw what the breaker handle says. Not at all, they're hardly ever loaded anywhere near that. In fact, they're rarely loaded at all. Power companies' rule of thumb is the average house draws 1KW on average - really. 4 amps @ 240V. (8 amps @ 120V).
But don't take my word on it, do a Load Calculation on the non-240V loads.
So, this suggests a simpler and cheaper plan: Leave the large 240V loads in the original 200A panel. There aren't very many of them. Have the new panel be for "anything/everything else". 65A would be plenty; 100A would be overkill; 120A would be gross overkill. All those breaker sizes are affordable compared to 200A. We don't get the 310.15(B)(7) derate*, so 120A takes #1/0 wire, 100A takes #1 wire and 65A takes #4 wire. (we're talking aluminum here). 65A and 120A breakers aren't made, so round up to the next available size (70A and 125A).
The gory details on 310.15(B)(7)
310.15(B)(7) 120/240-Volt, Single-Phase Dwelling Services and Feeders. For one-family dwellings and the individual dwe11ing units of two-family and multifamily dwellings, service and feeder conductors supplied by a single-phase, 120/240-volt system shall be permitted be sized in accordance with 31 0.15(B)(7)(l) through (4).
(l) For a service rated 100 through 400 A, the service conductors supplying the entire load associated with a one-family dwelling. or the service conductors supplying the entire load associated with an individual dwelling unit in a two-family or multifamily dwelling, shall be permitted to have an ampacity not less than 83 percent of the service rating.
(3) In no case shall a feeder for an individual dwelling unit be required to have an ampacity greater than that specified in 310.15(8)(7)(1) or (2).
83% of 200A is 166A. The smallest wire that will cover that is 4/0Al at 180A. That's why we pick it.
Since the service wire from (B)(7)(1) only needs 166a ampacity, (B)(7)(3) says the same is true of the feeder, so it can also be 4/0.
* Well, we do get the derate from 310.15(B)(7)(3), but it only gives us the ability to use 4/0 for that 100A feeder, not #2.