Go ahead and attach to the bottom lugs.
I am confused as to whether you have 125A or 200A service. No matter; the answer is the same way.
The wire "you happen to have lying around" is correct service wire for 200A. How do we arrive at that when the book plainly says 180A? Using NEC 310.15(B)(7) which says service wires to a dwelling only need to have ampacity for 83% of the service, or 166A for a 200A service.
You are using it for feeder to a subpanel. Does 310.15(B)(7) still apply? Well it doesn't apply to feeders generally, we can't take the 83% discount on any old feeder. However, it does say feeder wire never needs to be larger than the service wire needs to be. So if 4/0 is legal for a 200A service, it is legal for any feeder coming off a 200A service.
"That was easy"
"Wait. 4/0 wire says it's good for 205A @ 90C thermal. Aren't we good?" No, that only works if the terminals and enclosures on both ends are rated for 90C thermal.
"But wait. Won't that make it a 200A feeder?"
Yup. Yup, it certainly will.
That means you'll need to use a 200A rated subpanel.
But we would have you use a 200A subpanel anyway. Because the #1 panel problem we see around here is "Help, I am out of breaker spaces, how do I fit this new thing in?" And why did that happen? The guy who bought the panel was short-sighted and thought "Oh, no one will need more than 640K RAM!" (oh wait, that's computers.) And 125A can power a LOT more stuff than you might think.
Yes, at the time you buy the panel (e.g. now), you are 100% certain this is everything you will ever need and you will never want another thing. But then you do.
At the end of the day, spaces are cheap. We'd have you buy a 30-40 space panel just for plenty of spaces. Almost all panels that size are 200A.
Note that "12 space/24 circuit" is baloney! Due the modern rules, you rarely get a chance to use double-stuff breakers anymore, so that 24-circuit number is unattainable. Spaces will also be essential to accommodate the emerging "smart breaker" tech which will let you take advantage of new rate plans, and use things like solar or generator more intelligently.
You could use a subpanel with a main breaker smaller than 200A, but you're saving very little money, and you must still find a smaller breaker capable of taking 4/0 wire. Also, you pointlessly limit how much power you can take from the feeder at the garage.