I am not sure you can avoid the concept of Palm probabilities, since the conditional expectation you need for the mad king problem is precisely what Palm probabilities are designed for. If I understand correctly your generalized definition of a conditional expectation, the formula you give ($\int_\Omega\ldots$) is the exact analog of the formula I gave in my answer for the definition of the Palm probabilities. Let me suggest you some references: Stoyan, Kendall and Mecke: "Stochastic geometry and its applications" Scheider and Weil: "Stochastic and Integal Geometry".

Indeed, the graph I proposed is homogeneous: Each node of level n has exactly k^n paths to the root of the diagram. But then I must admit that this poses some problem, since this introduces eigenvalues for the associated adic transformation. So this adic transformation is rather an extension of the Bernoulli shift (probably a direct product with an odometer).