My take on Rita Mae's utter hate, nay contempt,for Billie Jean King, comes down to 2 facts I can be certain of and one wild hunch.
1. Brown hates closet gays for being dishonest. To her they inevitably become poisonous. She even went after Martina for this when Navratilova was going through her "I am bisexual not a lesbian" stage when she was living with Nancy Lieberman. King was as closeted as one could get.
2. Billie Jean had an obvious mental hold on Navratilova, who admired her greatly. At the very least Rita Mae saw this as a fight for Martina's soul, if you will. My memory may be bad, but it's possible Rita Mae had a hand in ending the King-Navratilova doubles partnership.
Wild speculation
Rita Mae knew, or thought she knew, that King and Navratilova had been former lovers. I have absolutely NO proof of that, but it makes sense. It explains Brown's insecurity regarding BJ, and the venom she pours forth sounds personal IMO. There is no doubt who Carmen Semana is in Brown's barely fictional book "Sudden Death". The bitchy jilted ex-lover Susan Reilly out for revenge is clearly Billie Jean. This is Rita Mae's not so subtle announcement that they were involved.