is it actually possible to use pure Bayesianism in any non-contrived, non-trivial context?

Sure. The difficulty of actually doing Solomonoff induction is irrelevant, because SI isn't actually part of Bayesianism as everyone in the world except Eliezer Yudowsky defines it. Cox's Theorem gives us the basic laws of probability, but there is nothing comparable telling us that algorithmic probability is the correct prior we should be using. A prior is an encoding of one's prior knowledge / state of information before seeing the experimental data, and we have no a priori reason to expect simple explanations for everything.