$\mathbb{F}_p((X))$ is decidable as a module over the ring of additive polynomials

Let $p$ be a prime number, $K$ be the henselization of the rational functions over the finite field $\mathbb{F}_p$ and $R$ be the ring of additive polynomials over K. We show that the field of Laurent series over $\mathbb{F}_p$ is decidable seen as an R-module. Moreover, we provide a recursively enumerable axiom system (satisfied by $K$) in the language of $R$-modules together with a unary predicate for the valuation ring, modulo which every positive primitive formula is equivalent to a universal formula. Consequently the $R$-module theory of the field of Laurent series is model-complete in this language and admits $K$ as its prime model.