Not only should you interconnect the panels with conduit (better than cable)... you should interconnect them with a number of conduits for convenience of pass-thru. You'll thank me later - I guarantee it!
Drill right through the studs. For the large pipe (1-1/2 or 2") I would use RMC the right length, with conduit nuts on the outside only. That way the hole doesn't need to be any bigger than the RMC. This is not good enough to carry ground, so run a ground wire.
Siemens designs their knockouts to a standard pattern, so KOs on any Siemens panel will line up with KOs on any other. Make cross-connections using conduit nipples or RMC conduit.
Make a 2" pipe for the feeder, and as many 3/4" cross-connections as you are willing to do, for future use by branch circuits.
If you are able to use 4/0 hots and 2/0 neutral, then you only need 1-1/2" conduit for the feeder.
Both hots and neutral must go through the same conduit. Ground can go another path, which is how you can make it work in 1-1/2" conduit.
You can run the whole SER through 2" conduit, or if the individual wires inside the SER have markings identifying their insulation type, they can be shucked out of the sheath and used as they are. Or you can use THHN or XHHW individual conductors.
Any branch circuit can be moved to the other panel as simple as moving its hot(s) and neutral via a 3/4" cross connection - the safety ground can stay in the panel the cable enters. Wires can be extended with wire nuts. They're allowed in the feeder cross-connection (space allowing), but I recommend several at least near top and bottom so you have the best chance of the existing wires reaching into the other panel to the new breaker.
As far as getting a 2/0 neutral connection, once we get you out of "fat cable" and into "individual conductors", you can simply use 2/0 copper for the neutral wire. It's only a few feet. Copper is perfectly safe at these large feeder sizes :)
I don't normally go around "spending your money", but in this case it's worth it for a few feet of 4/0AL and 2/0Cu and a much cleaner, pro tier installation that will be easy to manage and maintain (with all those pass-thrus).
Note that 4/0 is only legal for 200A if your entire service is 200A due to the 310.15(B)(7) exception. If this is half a 400A service, the exception does not apply, and 250 kcmil is required. (or #3/0 copper).