Subpanel for finished basement: sizing, cable, installation advice
That 1/0 SER is good for 125A, so you might as well use it for all its worth

A 1/0 Al SER cable can handle 120A when landed on 75&deg;C terminations, and we are allowed to round up to 125A by the 240.4(B) "round-up" rule as the next standard breaker size above 120A is 125A.  As a result of this, and the fact that a BR2125 is only $10-$15 more than a BR2100, along with the fact that the HOM2448L125PGC is a 125A panel to begin with (there is no 100A version in main lug), there is really no reason to go with anything less, although if you wanted to upsize the panel further, the HOM3060L225PGC is only $15 more.  (While you could run a 4/0 Al SER cable, you can't pull more than 125A out of your current panel anyway as the double frame breaker or subfeed lug required for a 200A feed takes up more space in your panel than you can make room for.)

Your grounding is good, but you'll need to remember that you'll need new breakers

While you're good to go on the one thing that often gets people in trouble with subpanels, namely grounding, as long as you keep your neutrals and grounds straight, thanks to the fact both panels mentioned above come from the factory with ground bars and a ground lug fitted already, you will have to remember that you'll need new Square-D HOMeline breakers for the circuits you're moving to your new panel, instead of being able to reuse the BR breakers you're pulling out of your existing panel.

TORQUE ALL LUGS TO SPEC

There's one more thing you'll need to do in order to ensure a reliable installation.  Your panel and breakers will have lug torque specifications labeled/marked on them (in inch pounds), and 110.14(D) in the 2017 NEC requires that you use a torque wrench or torque screwdriver to set the lug setscrews to that torque.  Even if your jurisdiction has not adopted the 2017 NEC yet, it's still a good idea anyway, since you don't want your electrical system to lose you the race, now, do you?