$\newcommand{\C}{\mathbb{C}}$ For $n=2$ it doesn't exist. For $n\geq 3$ it does. Let $P$ be any irreducible polynomial with trivial stabilizer in $G:=SL_n(\C)$. Let $K$ be the function field of the variety given by the equation $P=0$. Then there is a $K$- valued point $x_0$ in $\mathbb{P}^{n-1}$ with a trivial stabilizer. Let $X$ be the $G$- orbit of $x_0$ (just as a set). We can choose a local parameter at every point $x\in X$ in a $G$-equivariant way. Now every rational function $f$ can be expanded in every point $x \in X$ as a power series with coefficients say $a_k(f,x)\in K$. Take $\varphi(f)=\sum_{x\in X} a_{-1}(f,x)$. This is a non-trivial $K$- valued $G$-invariant functional on rational functions. Composing $\varphi$ with some $\C$-linear map $K\to \C$ will produce the desired functional.

EDIT: It shouldn't be a local parameter at every $x$, it should be $k-1$ of them. Anyway, one can expand rational functions in several variables into power series and extract coefficients, so one can construct lots of functionals like this.

$\newcommand{\C}{\mathbb{C}}\newcommand{\P}{\mathbb{P}}$ For $n=2$ it doesn't exist. For $n\geq 3$ it does. Let $P$ be any irreducible polynomial with trivial stabilizer in $G:=SL_n(\C)$. Let $K$ be the function field of the variety given by the equation $P=0$. Then there is a $K$- valued point $x_0$ in $\mathbb{P}^{n-1}$ with a trivial stabilizer. Let $X$ be the $G$- orbit of $x_0$ (just as a set). We can choose a local parameter at every point $x\in X$ in a $G$-equivariant way. Now every rational function $f$ can be expanded in every point $x \in X$ as a power series with coefficients say $a_k(f,x)\in K$. Take $\varphi(f)=\sum_{x\in X} a_{-1}(f,x)$. This is a non-trivial $K$- valued $G$-invariant functional on rational functions. Composing $\varphi$ with some $\C$-linear map $K\to \C$ will produce the desired functional.

EDIT: It shouldn't be a local parameter at every $x$, it should be $k-1$ of them. Anyway, one can expand rational functions in several variables into power series and extract coefficients, so one can construct lots of functionals like this.

$\newcommand{\C}{\mathbb{C}}$ For $n=2$ it doesn't exist. For $n\geq 3$ it does. Let $P$ be any irreducible polynomial with trivial stabilizer in $G:=SL_n(\C)$. Let $K$ be the function field of the variety given by the equation $P=0$. Then there is a $K$- valued point $x_0$ in $\mathbb{P}^{n-1}$ with a trivial stabilizer. Let $X$ be the $G$- orbit of $x_0$ (just as a set). We can choose a local parameter at every point $x\in X$ in a $G$-equivariant way. Now every rational function $f$ can be expanded in every point $x \in X$ as a power series with coefficients say $a_k(f,x)\in K$. Take $\varphi(f)=\sum_{x\in X} a_{-1}(f,x)$. This is a non-trivial $K$- valued $G$-invariant functional on rational functions. Composing $\varphi$ with some $\C$-linear map $K\to \C$ will produce the desired functional.

$\newcommand{\C}{\mathbb{C}}$ For $n=2$ it doesn't exist. For $n\geq 3$ it does. Let $P$ be any irreducible polynomial with trivial stabilizer in $G:=SL_n(\C)$. Let $K$ be the function field of the variety given by the equation $P=0$. Then there is a $K$- valued point $x_0$ in $\mathbb{P}^{n-1}$ with a trivial stabilizer. Let $X$ be the $G$- orbit of $x_0$ (just as a set). We can choose a local parameter at every point $x\in X$ in a $G$-equivariant way. Now every rational function $f$ can be expanded in every point $x \in X$ as a power series with coefficients say $a_k(f,x)\in K$. Take $\varphi(f)=\sum_{x\in X} a_{-1}(f,x)$. This is a non-trivial $K$- valued $G$-invariant functional on rational functions. Composing $\varphi$ with some $\C$-linear map $K\to \C$ will produce the desired functional.

EDIT: It shouldn't be a local parameter at every $x$, it should be $k-1$ of them. Anyway, one can expand rational functions in several variables into power series and extract coefficients, so one can construct lots of functionals like this.