Add sub panel from transfer switch
You can tap this

What you want to do here is replace the existing set of 3-way splice connectors that tap the feeder for the surge protector with 4-ways, and then use the newly opened connector-ports to run a feeder up to the transfer switch.

You're fine on size

The reason why the existing feeder is 2AWG is because it's copper most likely -- 2AWG copper is good for 115A at 75&deg;C, which is fine for a dwelling unit feeder application due to the 83% rule.  Your 1/0 aluminum wire will be fine, though -- the ampacity rating's about the same, and CuxAl rated Polaris connectors work fine for copper-to-aluminum connections when torqued to spec (AlumiConns are essentially miniature versions of them, even).

You will need a 125A subpanel for the standby loads given the style of transfer switch (DPDT safety switch) you linked -- I would recommend getting a 30 space main lug unit.  (Siemens has the advantage that the PL line of loadcenters comes with some niceties that you don't get with BR, but it seems like most of the supply houses in your area are loyal to Eaton, so a BR loadcenter is fine and dandy here.)

One caveat here

Since you got a two-pole transfer switch (instead of a three-pole), you must consult your generator manual (or generator manufacturer) for the correct procedure for pulling the neutral-to-ground bond on your generator, as your transfer equipment is set up for a non-separately-derived generator source.  Otherwise, you'll have issues (such as false GFCI trips on the generator with some generators, and energized boxes due to errant neutral current flowing on grounds) due to the duplicate neutral-to-ground bond on the generator.