Is "duplicate" in RPN enough for replacing variable binding in term expressions?
Is “duplicate” in RPN enough for replacing variable binding in term expressions?


No, it is not strong enough. One can eliminate $\mathsf{dup}$ by replacing each sequence of $\mathsf{dup}$ by a sequence of $x$ preceded by one $\mathsf{sto}(x)$ operation. Since the single variable "$x$" is enough to eliminate all $\mathsf{dup}$ operations, any term expression which requires more than one variable to be stored efficiently can't be expressed (efficiently) using only $\mathsf{dup}$.

If the $\mathsf{inv}$ operation is omitted, then this whole intended representation by a term expression can be seen as a special case of the directed acyclic graph representation of a polynomial used in arithmetic circuit complexity. So the RPN stack from the question turns out to be just a representation of a tree. Now a directed acyclic graph can't be "emulated easily" by a tree, and especially a non-planar DAG will be "too difficult" for the $\mathsf{dup}$ operation.

In order to add the $\mathsf{inv}$ operation again, note that the ring of polynomials in $x_1,\dots x_n$ over $\mathbb Z$ is just the free commutative ring with generators $x_1,\dots x_n$. The corresponding free algebra with an $\mathsf{inv}$ operation would be the free commutative regular ring with generators $x_1,\dots x_n$. To be faithful to the canonical representation of the free algebra of a variety (equationally defined collection of algebras), the $\mathsf{inc}$ operation should be replaced by $1$.