From my perspective, first you should just try to find a biophysically plausible model which contains anyons. Then you can test it, e.g. by examining optical and electrical properties of the microtubule.

People measure such properties already, so we may already have relevant data, even evidence. There is a guy in Japan (Anirban Bandyopadhyay) who claims to have measured all sorts of striking electrical properties of the microtubule. When he talks about theory, I just shake my head, but that doesn't tell us whether his data is good or bad.