Sub Panel using Sub Feed Lugs
Your panel's limit on stab ampacity only applies to double-stuff breakers

Your panel's label should read, among other things:


  SUM OF QT BREAKER RATING IS NOT TO EXCEED 110A PER BRANCH CIRCUIT BUS STAB


Note here that it says "QT" breaker, while regular Siemens breakers are type QP.  This is because this restriction only applies to Siemens tandem aka "double-stuff" or "cheater" breakers, which use a type QT label to distingush them from normal type QP breakers, or the type QNR double frame breakers that are used for breaker ratings above 125A.

As a result, I would go ahead and put the 150A type QNR breaker in the main panel in question, using 2/0 Al XHHW-2 for the hots and neutral with a 6AWG bare copper ground in 2" PVC conduit.  (If you're stuck with copper because of local rules, 1AWG is the correct gauge, by the way.)

GO BIG OR GO HOME

For the panel in the ADU, we know two things:


Spaces are cheap now compared to the cost of replacing a panel later because it filled up
We need a main shutoff in the panel since we are dealing with powering a separate structure, but that main shutoff doesn't need to be a breaker (and can't be selectively coordinated with the feeder breaker in the main panel anyway!), just a switch


Given this, I would put a 42-space, 200A panel in the ADU and call it a day; if you don't mind losing a couple spaces, you could even use another S4080B1200A there, provided the panel is living indoors of course.

TORQUE ALL LUGS TO SPEC

There is one final thing to note here, and that is that you will need to use an inch-pound torque wrench or torque screwdriver to torque all loadcenter and circuit-breaker lugs to the torques specified on the labeling.  This is required by the 2017 NEC in 110.14(D), and also is a good idea anyway, especially with aluminum wire, lest you want your subpanel install to be as reliable as Greg Biffle's infamous lugnuts.