Edit
Tsirelson space admits spreading models 1-equivalent to the unit vector basis of $\ell_1$. See the paper of Odell and Schlumprecht in JFA titled A problem about spreading models. So $\beta =2$ and yet reflexive.
Ignore the below my first answer.
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.