Do I need to upgrade to a 400A panel with solar?
Short answer is "no"

There are a couple ways of fixing this short of going to 400A busbars in the main panel, which are not exactly the cheapest thing in the world.  Read on for options, although you'll need to discuss both with your AHJ and the latter option (using the secondary disconnect on a suitably equipped meter-main for the EVs) with your utility as well.

One other caveat with most of this is you're best off with 32A EV chargers (vs 40A) to avoid starving the house for power too much -- luckily, 32A@240VAC is a standard circuit size in most of the rest of the world, so you can get EV chargers that size fairly readily.

Easy fix -- downgrade the 200A feeder breaker in the main panel to 125A

Given the rules of NEC 705.12(D)(2)(3)(c) and that the total overcurrent protection for the planned EV chargers won't exceed 100A, the easiest solution to this problem is to downgrade the main panel's feeder breaker to 125A, which brings you back into compliance with the rules.  This does mean that if both the Powerwalls are charging at their full 30A rate, your solar system isn't helping at all, and you're pulling significant load (more than 65A) on the house side, you could trip the feeder breaker.

If you are OK with starving the EV chargers for current somewhat, or your AHJ is willing to apply 240.4(B) rules to this situation, then you can use a 150A feeder breaker, but that limits you to 70A (without 240.4(B)) or 80A (with 240.4(B)) maximum breakering for the EV chargers, taken together.  (If you could put a 75A breaker in, that'd work, but 75A is not a standard breaker size.)

If that's not an acceptable state of affairs...

If the above situation is not serviceable, then there is another option, but it involves replacing your meter socket with a specific meter main (or meter breaker), which may or may not be more costly than the 400A panel.  In particular, the Eaton/Cooper 2M2RP provides a 200A continuous socket with two load taps: one for a 200A maximum main breaker, and a second provision for an auxiliary service disconnect.  However, the lugs on the meter socket are limited to 125% of 200A (their continuous rating), or a 250A maximum, so the disconnects must sum to that (vs. the 225A maximum rating on your proposed main panelboard's busbars).

This means that you can use a maximum of a 175A main breaker in the unit (downbreakering it from the shipped 200A), and the auxiliary disconnect paired with that can be a maximum of 75A as a result; however, 75A isn't a standard breaker rating, so we can use an 80A breaker instead as per 240.4(B) (it's a feeder, we're protecting a set of lugs, and we're below the 800A maximum for the rule).  Given that the 2M2RP takes Siemens breakers (for historical reasons), and that Siemens has replaced the QJ line with the backwards-compatible QR, we need a QR22B175 for the main, and the provision for the secondary main can be left unpopulated for now.  When the EV chargers are installed in this scenario, a Q280 is inserted in that secondary main slot, and then a feeder is run from it to a main lug panel with the EV-charger branch breakers in it.

If your AHJ or utility rejects the application of 240.4(B) here, one can use a QR22B150 for the main and a Q2100 for the secondary, instead -- this does mean that pulling full power from the house while the Powerwalls are charging at full rate and the solar system is outputting nothing puts you in main-breaker-trip land, but that is quite unlikely.  Another option would be to use a QR22B175 for the main and a Q270 for the secondary, but that starves the EV chargers for current ever-so-slightly.