Fortunately, LLM queries are relatively expensive and time-consuming. If 1 query costs $0.01 and takes 1 millisecond, that may not seem like a lot. But if you're trying to DOS a server you'll probably need >1000 queries a second, and even if that's enough, such a DOS will cost you $36,000/hr. In this case, whether the captcha is solvable by the LLM doesn't matter.
I think it sends a crypto puzzle to your computer and when your computer solves it you're authenticated (their website isn't super clear).
Depending on your AI bot, it might not be hooked up to a cpu beyond just input > LLM > output, it would stop that