  We consider the stochastic inviscid Leray-$\alpha$ model on the torus driven
by transport noise. Under a suitable scaling of the noise, we prove that the
weak solutions converge, in some negative Sobolev spaces, to the unique
solution of the deterministic viscous Leray-$\alpha$ model. This implies that
transport noise regularizes the inviscid Leray-$\alpha$ model so that it enjoys
approximate weak uniqueness. Interpreting such limit result as a law of large
numbers, we study the underlying central limit theorem and provide an explicit
convergence rate.

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