A causal reasoner will compute about P(utility=U| do{action=$5}), which doesn't run into this trouble. This is the approach I recommend.

Probabilistic reasoning about actions that you will make is, to the best of my knowledge, not a seriously considered approach to making decisions outside of the context of mixed strategies in game theory, and even there it doesn't apply that strong, as you can see mixed strategies as putting forth a certain (but parameterized) action whose outcome is subject to uncertainty.

I don't think your sketch is correct for two reasons:

1. The assumption that your action is utility-maximizing requires that you choose the best action, and so using it to justify your choice of action leads to circularity.
2. Your argument hinges on P(U($10)>U($5)|A=$10) > P(U($5)>U($10)|A=$5), which seems like an odd statement to me. If you take the actions maximize utility assumption seriously, both of those are 1, and thus the first can't be higher than the second. If you view the actions as not at all informative about the preference probabilities, then you're just repeating your prior. If the action gives some information, there's no reason for the information to be symmetric- you can easily construct a 2x2 matrix example where the reverse inequality holds (that if we know they picked $5, they are more likely to prefer $5 to $10 than someone who picked $10 is to prefer $10 to $5, even though most people prefer $10 to $5.