i am currently solving a nonlinear PDE of mixed parabolic/hyperbolic type of the Form

\begin{align*} \frac{\partial}{\partial x} \left(A \frac{\partial p}{\partial x} \right) +\frac{\partial}{\partial y} \left(B \frac{\partial p}{\partial y} \right)=\frac{\partial F(p)H}{\partial t} \end{align*}

which is solved for p with the finite Volume Method. $A,B$ and $H$ are known. This results in a large system of equations \begin{align} \underline{\underline{A}}\, \underline{ p} = \underline{R}(p) \end{align} As you can see, the right hand side is dependent on p. The nonlinearity is adressed with a fixed point iteration or Newton-Raphson Iteration. The $F-p$ Curve looks like a logistic function https://en.wikipedia.org/wiki/Logistic_function

However, a regular fixed point iteration does not work, simply calculating new right hand sides and substituting them on the RHS will not result in a convergent solution. A working solution strategy is substituting $F(p)$ with \begin{align} F(p)=\Pi p\end{align} The newly introduced variable is a prefactor of $p$, so it can be dragged to the left hand side into the system matrix. So now the system matrix is dependent on $p$. This solution converges nicely, but of course at the cost of a nonlinear system matrix which makes the computation quite expensive. So my question is: Why does this converge, but the nonlinear RHS does not? And are there other solution strategies that do not need a nonlinear system matrix?

Edit: $\Pi$ is simply calculated by dividing the last calculated $F$ by the last calculated $p$