You are right - bio majors are more rare, and so may have a better chance of getting in. The same is true when you are applying for jobs - you will stand out among applicants, especially at firms that do intellectual property law.
It also opens up more possibilities of practice areas themselves. As a classics major with a law degree, you are qualified to go into any type of law except IP. As a bio major, you are qualified to do any area of law *including* IP.
The work itself involves reading a lot, mostly case law. Law school teaches you to think a certain way that is unique to law school While certain majors may emphasize critical thinking skills, this will only give you a very slight head start to the type of thinking you will need to do at law school.
And while the law uses latin terms, this is only occasionally. Knowing Latin will not help you much, as the few terms they use are easy to learn as you go.
The only reason I might go with classics is that the grading is easier and you will likely have a higher GPA. However, since most law schools take into account your major when looking at grades, I am not sure how much impact this has - especially if you're good at biology.
I think your instincts are correct on this one!
Law student, (with a firm job for after graduation) with friends at school who were classics majors, and friends who were bio majors.