And since a non-random cause for mutagenesis doesn't eliminate a role for natural selection, wouldn't natural selection encourage genetic engines more capable of utilizing this mutagenic fuel?
It did! Not knowledgable enough to give you the details, but I believe that are certain well defined portions or domains of DNA utilized in the creation of elements of the immune system which do mutate at significantly increased rates, and that this is part of a mechanism allowing the system to more rapidly create antibodies that bind strongly to foreign cells or particles.