I'm trying to solve the equation $$ \left(xy + \frac1e\right)e^{F(x,y)} = F(g(y)x , f(y)) $$ for $F(x,y)$ where $f$ and $g$ are known functions, analytic in a neighborhood of $0$, with $g(0) = 1$ , $f(0) = 0$, and $f'(0) = 1$. I know how to find a formal power series $$ F(x,y) = \sum_{n,m\ge0} a_{n,m} x^n y^m $$ but this is a hassle to do and I don't that it will converge for any nonzero $x,y$, so I'm wondering if there's an alternative way to solve this type of equation.
For my particular application, $f$ can be any analytic function satisfying $f(0) = 0$ and $f'(0) = 1$, but $g$ will be related to $f$ by $f(y) g(y) = y$.