This paper extends our prior work on "E-type" (exponential family type) channels. The channels considered here have transition kernels induced by an exponential family with a two-component sufficient statistic composed of an input-output distortion function and an output cost function. We demonstrate the existence of a mutual information saddle point in any E-type channel for which there exists a source distribution such that the induced output distribution is maximum-entropy under an output cost constraint. For additive-noise E-type channels, we provide necessary and sufficient conditions on the existence of saddle points which coincide with convolution divisibility of the additive noise law. This machinery generalizes many well-known saddle-point, capacity, and rate-distortion theorems, including those for the additive Gaussian and exponential-noise channels, and leads to a saddle point result on the non-additive exponential server timing channel, which appears to be new.