In order to mess with anthropic probabilities in isolation you'd need to change the structure of coinflips and people-creation
But you can't mess with the probabilities in isolation. Suppose I were an SIA agent, for instance; then you can't change my anthropic probabilities without changing non-anthropic facts about the world.
I'm uncertain whether what you're saying is relevant. The question at hand is, is there some change to a problem that changes anthropic probabilities, but is guaranteed not to change ADT decisions? Such a change would have to conserve the number of worlds, the number of people in each world, the possible utilities, and the "no anthropics" probabilities
For example, if my anthropic knowledge says that I'm an agent at a specific point in time, a change in how long Sleeping Beauty stays awake in different "worlds" will change how likely I am to find myself there overall.
A few weeks ago at a Seattle LW meetup, we were discussing the Sleeping Beauty problem and the Doomsday argument. We talked about how framing Sleeping Beauty problem as a decision problem basically solves it and then got the idea of using same heuristic on the Doomsday problem. I think you would need to specify more about the Doomsday setup than is usually done to do this.
We didn't spend a lot of time on it, but it got me thinking: Are there papers on trying to gain insight into the Doomsday problem and other anthropic reasoning problems by framing them as decision problems? I'm surprised I haven't seen this approach talked about here before. The idea seems relatively simple, so perhaps there is some major problem that I'm not seeing.