When I applied for my current position as a paralegal for a biglaw firm, I hadn't taken the LSAT yet, so I wasn't in your situation, but I don't think I would have included it if I had. Throughout the hiring process, all of my communication was with HR, and I did not get the impression the HR people would care what my LSAT score was or know how to determine what a "good" LSAT score is. Once I got my interviews with firms, I spoke to other paralegals and attorneys at the firm and was asked several times if I had taken the LSAT yet. I think that would be the more appropriate time to let them know, and I would have gladly shared my score with them then if I had had it.
As for the advice about not mentioning you want to go to law school, that is really bad advice and you should disregard it. I interviewed with four v100 firms and each one of them asked me if I was considering law school and mentioned that they typically try to hire paralegals that stay for two years and then go onto law school. They know it's a revolving door, and they want candidates that are interested in the work but won't be staying long. Once I started at my firm, I was completely upfront with my coworkers and the attorneys and was able to approach them for advice. One associate wrote me a LOR, another associate tutored me for the LSAT, another associate helped me with my resume, and the firm's resident legal writer proofread my PS. Being a paralegal was a great experience for me. I was able to write about it in my PS, and on several of my acceptance letters, the deans' handwritten notes mentioned my paralegal experience as valuable.
I think it's too soon for you to be giving up hope for the cycle, but if you do decide to be a paralegal, it can definitely strengthen your application.