Is the Landau free energy scale-invariant at the critical point?

I answered a very similar question here, but in the context of quantum field theory rather than statistical field theory. The point is that it is impossible to have a nontrivial fixed point classically (i.e. without accounting for quantum/thermal fluctuations) for exactly the reason you stated: the dimensionful coefficients will define scales.

We already know that quantum/thermal fluctuations can break scale invariance, e.g. through the phenomenon of dimensional transmutation, where a quantum theory acquires a mass scale which wasn't present classically. And what's going on here is just the same process in reverse: at a nontrivial critical point the classical scale-dependence of the dimensionful coefficients is exactly canceled by quantum/thermal effects. Of course this cancellation is very special, which is why critical points are rare.