Can a subpanel's ground and neutral be served from another subpanel, while the circuits come from main panel?
I'd simply run the new neutral back to the "main" panel

While adding accessory ground bars to your "main" panel will be trickier than it looks, since you'd have to field-fabricate a replacement for the neutral tie strap that originally came with said panel in addition to unbonding the left-hand bar and moving all its ground wires onto a newly installed ground bar, you do have enough neutral space on the right-hand bar for an ECLK1-2 lug to fit there, and you can do the same with the left-hand bar for your new grounding wire.

From there, you can run the wires from the new cable through the subpanel and existing nipple to reach the "main" panel, as there's no shortage of space in that nipple for more wiring.  You will need to label on the subpanel's directory that there are feed-through conductors present and their disconnecting means is in the "main" panel next to said subpanel, as well, as this is required by NEC 312.8(A) point 3.

The alternative plan you propose of landing the neutral in the subpanel and continuing the hots off to the main panel is legal, under a little-used Code provision found in NEC 215.4:


  215.4 Feeders with Common Neutral Conductor.
  
  (A) Feeders with Common Neutral. Up to three sets of 3-wire
  feeders or two sets of 4-wire or 5-wire feeders shall be permitted
  to utilize a common neutral.
  
  (B) In Metal Raceway or Enclosure. Where installed in a
  metal raceway or other metal enclosure, all conductors of all
  feeders using a common neutral conductor shall be enclosed
  within the same raceway or other enclosure as required in
  300.20.


But, given the short distance of the shared run and the fact that there is space in the neutral bar for another add-a-lug, there's really not that much point in finessing things that tightly (and risking confusing an inspector who doesn't see that sort of thing on a regular basis at all).

HOWEVER: you'll need either a 90A breaker or 1-1-1-3 cable

There is a problem with your plan, though: namely, that 2-2-2-4 Al SER cable you are planning on using is only rated for 90A at 75&deg;C (which you are allowed to use when running from panel to panel).  So, you'll either need to get a Q290 for your feeder breaker instead of a Q2100, or run 1-1-1-3 Al SER instead of the 2-2-2-4.

While you're in there, you'll want to swap the two breakers in the far bottom left of the "main" for the correct Siemens breakers, since the existing breakers there are foreign to your panel, and thus won't fit quite right.  In particular, the BR250 in the far bottom left position of that panel needs to be swapped out for a Q250, and the mystery 40A breaker above it needs to be swapped out for a Q240.

Other notes

It would be a good idea to replace the existing subpanel with something with more spaces; given the room available in that bay, using a 200A, main lug Siemens panel with 40, 42, or even 54 spaces for the replacement subpanel would not be at all out of place.  I would also use a larger subpanel than your existing 20-space panel for the new addition; a 30-space, 125A, main lug subpanel would not be out of place there, either.

Also, as with any panel work, you'll want to use an inch-pound torque wrench or torque screwdriver when tightening the lug setscrews on your connections.  Not only is this a NEC requirement as of 2017, found in 110.14(D), it's a very good idea anyway, lest your electrical system lose you the race!