Interesting! Thank you for writing this, this is something I was also wondering about while reading for the Warwick EA fellowship. My intuition is also that in the case of a "many-membered set of probability functions", I'd define a prior over those and then compute an expected value as if nothing happened. I acknowledge that there is substantial (or even overwhelming) uncertainty sometimes and I can understand the impulse of wanting a separate conceptual handle for that. But it's still "decision making under uncertainty" and should therefore be subsumable under Bayesianism.
I feel similar to ben.smith that I might be completely missing something. But I also wonder if this confusion might just be an echo of the age-old Bayesianism vs Frequentism debate, where people have different intuition about whether priors over probability distributions are a-ok.