Here is a *real* solution for $k=2$: Take $f_1(x)=x$ and
$$f_2(x)=\arccos\left( \sqrt{\tfrac{3}{4}- \cos ^4(x/2)} \operatorname{cotan} (x/2)-\tfrac{1}{2}\cos x\right).$$
Then
$$\cos f_1(x)+\cos f_2(x)=\cos\bigl(f_1(x)+f_2(x)\bigr),\;\;\text{for}\;\;\pi/3<x<\pi +2 \arcsin \left(\frac{3^{1/4}}{\sqrt{2}}\right).$$
Numerically, this interval is $1.05<x<5.53$.

This can be generalized to larger $k$; put all $f_n(x)=x$ for $n=1,2,\ldots k$ and then solve for $f_k(x)$.