I know I'm very late to this, but definitely she is autistic, I think. The person she's based on, Tennessee's older sister Rose, was diagnosed schizophrenic, but back then, autism was very unheard of and was often mistaken for schizophrenia. I can't find anything that said Rose had delusions and hallucinations, which is probably the main definer for schizophrenia, so autism would probably be a more likely and accurate diagnosis.
Also I feel too much like Laura in real life. I've basically sort of become her, including being fond of a guy named Jim in high school, after seeing one of my best friends perform as her at my college. I really wanted the part, but she still did wonderfully.