Edit
Tsirelson space admits spreading models 1-equivalent to the unit vector basis of $\ell_1$. See the paper
Odell, E.; Schlumprecht, Th. A problem on spreading models. J. Funct. Anal. 153 (1998), no. 2, 249--261. MR1614578
So $\beta =2$ and yet reflexive.
Ignore my first answer below.
If you allow renormings of the space when computing $\beta$ then $\beta(S_X)=2$ implies that $X$ is non-reflexive. This follows (I think) from a theorem of Odell and Schlumprecht.
