One cannot get into an Ivy league school with only a “good” LSAT score. It has to be close to the top, as does one’s GPA. However, from what I have read BTLs do very well on the LSAT as it tests analytical skills (BTW, you can see the ranking of majors by average LSAT scores online and you will see that Pe-Law is not very high). Moreover, many schools look for a diversity of majors.
Having written that, I do not understand why anyone who is not rich or good enough to get a full scholarship would go to law school today. Tuition is out of control (about five times the inflation rate over the last forty years – to give an idea, the gross starting salary at a NYC DA office forty years was twice the total three year tuition whereas now it is only a bit over one year’s tuition) and, as CTL observed, much work that was done by lawyers can be done by lay people (or paralegals). If someone likes the law but is not rich or a genius I would advise taking a paralegal course at a reputable school. It is much cheaper and faster.