I wasn't able to find the analytical solution but I at least tried to make a few steps in the right direction, which was enough for a B from the professor, which likely means it was indeed a step in the correct direction.
The solution of a 2D wave equation with a source is a sum of homogenous + particular solution of the equation. $$ w(x,y,t) = w_h(x,y,t) + w_p(x,y,t)$$ The boundary conditions for the system at the air-membrane junction are $$\dot{\xi}_{air}((L-w(x,y,t),t) = \frac{\partial w(x,y,t)}{\partial t}$$ and $$p(x,y,t) = p_{air}(L-w(x,y,t),t)$$
The obstacles in being able to solve this equation with given boundary conditions are
i) $L-w(x,y,t)$$(L-w(x,y,t))$ is problematic. The approximation of $p_{air}(L-w(x,y,t),t) = p_{air}(L,t)$ is not feasible.
ii) The necessity to find a particular solution of a sourced 2D wave equation satisfying these conditions.
Due to my background in Mechanical engineering (and lack of experience with finding solution of this type of PDEs) I wasn't able to continue any further.
If anyone ever faces a similar problem and manages to make any further progress, please do contact me as I'm interested in what the actual solution looks like.