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.][1]


  [1]: http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00251-3/S0894-0347-98-00251-3.pdf