I don't think this can be done without addressing the issue of infinite sums in $k$. If we have such a mapping $\varphi:k[[x_1,\cdots,x_m]]\rightarrow k[[y_1,\cdots,y_n]]$ with
$\varphi(x_i)=a+g(y_1,\cdots,y_m)\in k[[y_1,\cdots,y_m]]$ where $a\in K$, $a\not=0$, and $g(0,0,\cdots,0)=0$, then let $f=1+x_i + x_i^2 + x_i^3 + \cdots$
What is $\varphi(f)$? If it's something even close to obvious, then it's the power series that one obtains by substituting $\varphi(x_i)$ into each term. However, this gives us a constant term of $1+a+a^2 + a^3 + \cdots$ which is something that we know about if, say, $k\in \{\mathbb{Q},\mathbb{R},\mathbb{C}\}$, but I'm not aware of any kind of study of infinite series in general fields (off the top of my head, I don't think that a series whose terms aren't eventually zero could possibly converge in a finite field).
And even if, say $k=\mathbb{R}$ and, say $a=\frac{1}{2}$, then the constant term of $\varphi(f)$ would be 2, but then consider what $\varphi$ would do to the power series $1+3 x_i + 9x_i^2 + 27 x_i^3 + \cdots + 3^n x_i^n + \cdots$ In this case, the series formed by the constant term diverges. What then?
The issue here is why we have the word "formal" in the phrase "rings of formal power series": in a general algebraic setting, we don't care about convergence, and we don't plug anything into our power series except zero.