After having received the answer, I did some googling work and found a proof of the existence of a Hamel basis which is also a Bernstein set on Nonmeasurable Sets and Functions, page 39, Theorem 4. The link is to the google book, but I cannot understand the proof. There seems no evidence that $F_\xi\setminus T_\xi\neq\emptyset$ from the fact that $\lvert T_\xi\rvert<c$ without CH. I looked up another proof of the fact that there's a nonmeasurable Hamel basis on Measure Theory, page 86, problem 1.12.66, which sharpens the condition that the compact sets should be of positive measure.
It seems to me that either the definition of Bernstein's set is wrong, or the proof of the first book is wrong. I need a resolution of this paradox.
Thanks!
I've found that I was ignorant, since every closed set of a Polish space contains a perfect subset, and a nonempty perfect subset need to be of cardinality $\mathfrak c=2^{\aleph_0}$.