Thanks for your answer! However I have one doubt left: it seems to me that you are assuming that the theory at the fixed point is a CFT. I know that it is a standard result, but I have never found a proof of that either, only some qualitative arguments. I asked about scale invariance because I feel that proving that a theory is scale invariant could be easier than proving it is invariant under the whole conformal group. However, if you also happen to have references for $\beta = 0 \Rightarrow $ CFT I would be equally satisfied!