Assume I have two independent random variables $X$ and $Y$ with distributions $F_X$ and $F_Y$ respectively. Moreover, I know that $F_Y= g(F_X)$ where $g(.)$ is a strictly increasing bijective function.
Is there an invertible function $h(.)$ linking the characteristic functions $\phi_X$ and $\phi_Y$? If so, how would I write in terms of $g(.)$ and $F$s?
No.
Note that your hypothesis $F_Y = g(F_X)$ will be satisfied whenever $F_X, F_Y$ are both strictly increasing (simply take $g = F_Y \circ F_X^{-1}$). So let's take $X \sim N(0,1)$, $Y \sim N(1,1)$. Then $\phi_X(t) = e^{-t^2/2}$, $\phi_Y(t) = e^{it-t^2/2}$. We have $\phi_X(1)=e^{-1/2} = \phi_X(-1)$ but $\phi_Y(1) = e^{i-1/2} \ne e^{-i-1/2} = \phi_Y(-1)$. So it is impossible to have $\phi_Y = h(\phi_X)$.
