OK I looked at the only piece of SER I have and it is 2-2-2-4 (2ga with a 4ga EGC) that exceeds 250.66 and assuming 2ga is worth 100a 250.122. You can't even get to #4 using 310.15 (B)(6) which lets you use a 125a breaker on your #2 and 250.122.
Looking at the Southwire site, that oversizing continues up through the larger sizes. For a 1/0 they sell you a #2 EGC and even with the 175a breaker you can use for that residential main feeder only wants a #6 as would 250.66.
https://www.mysouthwire.com/medias/...pecifications/he6/h1f/8854083436574.pdf.I think you are more than safe
