Preliminaries: Let $X$ be a projective smooth curve (scheme of finite type, integral and of dimension $1$) over a perfect field $F$. Let $K$ be the function field of $X$ and for a closed point $P$ let $K_P$ be the completion of $K$ with respect to the discrete valuation $v_P$ at $\mathcal O_{X,P}$.

By following "Tate - differential and residues on curves" one can define in a very abstract way the residue at a closed point $P$ as a map:

$$\text{res}_P:\Omega^1_{K|F}\rightarrow F$$ and moreover it is not difficult to prove the following properties :

1. Since $F$ is perfect, $\Omega^1_{K|F}$ has dimension $1$ over $K$. Therefore by chosing an uniformizer parameter $t\in\mathcal O_{X,P}$ for $v_P$ it is possible to write every non-zero differential form as $\omega=fdt$ where $f\in K^\times$

2. Let $\omega=fdt$, then $\text{res}_P(\omega)=\text{Tr}_{k(P)|F}(a_{f,-1})$ where: $a_{f,-1}$ is the coefficient of $t^{-1}$ in the expansion of $f$ as Laurent power series and $k(P)$ is the residue field at $P$.

Is it possible to define a (complete) residue map: $$\text{res}_P:\Omega^1_{K_P|F}\rightarrow F$$

such that the properties $1.$ and $2.$ hold? I'd like to have a representation $\omega=gdt$ with $g\in K_P$ and $\text{res}_P(\omega)=\text{Tr}_{k(P)|F}(a_{g,-1})$.

It seems that the key point is that $\Omega^1_{K|F}$ is one dimensional, but what about $\Omega^1_{K_P|F}$?