Does this "inverse Taylor series" exist in literature?

Suppose that you write $$x=\sum_{k=-n}^{k=+n} c_k\, e^{k x}$$ instead of Taylor expansions, consider the norm $$\Phi_n=\int_{-a}^{+a}\Bigg[x-\sum_{k=-n}^{k=+n} c_k\, e^{k x}\Bigg]^2\,dx$$ and minimize it with respect to the $c_k$'s. If we use a symmetric interval, we shall have $c_{k}=-c_{-k}$ and $c_0=0$. So, for the example you gave, consider $a=1$ and $n=2$ and the minimization will lead to $$c_{-2}=-\frac{3 e^2 \left(7-84 e^2+54 e^4-28 e^6+3 e^8\right)}{1+84 e^2-57 e^4+448 e^6-201 e^8+12 e^{10}+e^{12}}$$ $$c_{-1}=\frac{6 e \left(3+9 e^2+38 e^4+6 e^6-9 e^8+e^{10}\right)}{1+84 e^2-57 e^4+448 e^6-201 e^8+12 e^{10}+e^{12}}$$ $$\Phi_2=\frac{3 \left(15+324 e^2+134 e^4+804 e^6-405 e^8+40 e^{10}\right)}{1+84 e^2-57 e^4+448 e^6-201 e^8+12 e^{10}+e^{12}}-\frac{23}{6}=2.75\times 10^{-6}$$ As usual, the maximum error is at the bounds; in this case, it is $3.83\times 10^{-3}$.

Using $n=3$, the norm decreases to $7.37\times 10^{-9}$ (a factor of $373$)

Edit

Using as you did the series expansion $$e^{k x}=1+k x+\frac{k^2 x^2}{2}+\frac{k^3 x^3}{6}+\frac{k^4 x^4}{24}+O\left(x^5\right)$$ and using the same norm to minimize the error, I find $$x=\frac 1{12}e^{-2x}-\frac 23 e^{-x}+\frac 23 e^{x}-\frac 1{12}e^{2x}=\frac{1}{3} \sinh (x) (4-\cosh (x))$$

Here is a plot with the two approximations (black: minimizer of the norm with exponential functions; red: minimizer of the norm using polynomial expansion of $e^{kx}$)

Update

Taking into account the above, it means that we make the approximation $$x_{(n)}=\sum_{k=1}^n c_k \, \sinh(k x)$$ and the parameters are obtained by the minimization of the norm $$\Phi_n=\int_{-a}^{+a}\Bigg[x-\sum_{k=1}^n c_k \, \sinh(k x)\Bigg]^2\,dx$$ where the hyperbolic sines are either used per se or approximated by their Taylor series; the last solution is obviously simpler in terms of apparent "simplicity" of the coefficients.

However, for a given $n$, the results will depend on the number of terms used in the Taylor series. Expanding the hyperbolic sines to $O\left(x^{2p+1}\right)$, the coefficients are (still for $a=1$) $$\left( \begin{array}{ccc} p & c_1 & c_2 \\ 2 & 1.333333333 & -0.1666666667 \\ 3 & 1.266696757 & -0.1365320137 \\ 4 & 1.260198645 & -0.1337118902 \\ 5 & 1.259839496 & -0.1335590439 \\ \cdots &\cdots &\cdots \\ \infty &1.259826765 &-0.1335536879 \end{array} \right)$$