Suppose we have a Lawvere theory $L$, i.e. a category with finite products and objects $[n]$ given by the natural numbers such that $[n] \cong [ 1 ] ^n $, and localize it to a Lawvere theory $S^{-1}L$ by inverting an arrow in it (this works: one can e.g. get a small monoidal category with the arrow inverted as in this question, then make it into a category with finite products as in this question).

Now I consider models in $Set$ of the theories so obtained:

The morphism $L \to S^{-1}L$ induces a forgetful functor $S^{-1}L-Alg \to L-Alg$ which has a left adjoint (given by left Kan extension). Analogously to the case of localization of rings this left adjoint preserves finite limits, if our arrow had domain and codomain $[ 1 ]$, i.e. we just inverted an operation with one input and one output. My question is this:

Does anybody know conditions under which the inversion of more general arrows also induces a finite limit preserving left adjoint? Simple examples are welcome. What about the free Lawvere theory on one operation $[n] \to [n]$ which we then invert? Most importantly: What about the free commutative such Lawvere theory?