If you want to compute a Hausman test statistic that works also with cluster-robust standard errors you can follow the procedure outlined in Wooldridge (2010) "Econometric Analysis of Cross-Section and Panel Data".
Let $x$ include all time-varying variables. You need to compute the random effects differences for the dependent $y_{it} - \widehat{\theta}\overline{y}_i$ and explanatory variables, $x_{it} - \widehat{\theta}\overline{x}_i$, as well as the within transformed explanatory variables $x_{it} - \overline{x}_i$. Then you can estimate the OLS regression
$$y_{it} - \widehat{\theta}\overline{y}_i = (1-\widehat{\theta})\alpha + (x_{it} - \widehat{\theta}\overline{x}_i)'\beta + (x_{it} - \overline{x}_i)'\gamma + \epsilon_{it} $$
The robust Hausman test amounts to a Wald test for $H_0:\gamma =0$. This is asymptotically equivalent to the standard test if random effects without clustered errors is already efficient.
In terms of programming this is easy if you have a balanced panel. If not, then this complicates things in the sense that you need to estimate $\widehat{\theta}_i$ for every panel unit. If you are interested in the Stata code you can have a look at Cameron and Trivedi (2009) "Microeconometrics Using Stata".