I like the formalization, but it seems to miss a key feature of the parliamentary model. Per Bostrom,

[...] even a relatively weak theory can still get its way on some issues that the theory think are extremely important by sacrificing its influence on other issues that other theories deem more important. For example, suppose you assign 10% probability to total utilitarianism and 90% to moral egoism (just to illustrate the principle). Then the Parliament would mostly take actions that maximize egoistic satisfaction; however it would make some concessions to utilitarianism on issues that utilitarianism thinks is especially important. In this example, the person might donate some portion of their income to existential risks research and otherwise live completely selfishly.

If preferences are only defined by an ordering of possible outcomes, then you would get something like this:

• Total Utilitarian := (Donate 100% of income to existential risk reduction and otherwise behave selflessly, Donate 100% to x-risk and behave egoistically, Donate 40% and behave selflessly, Donate 40% and behave egoistically, 0% and selfless, 0% and egoistic)

• Egoist := Reverse(Total Utilitarian)

Then what particular reason do we have to expect them to end up compromising at [40% and egoistic], rather than (say) [0% and selfless]? Obviously the total utilitarian would much prefer to donate 40% of their income to x-risk reduction and behave selfishly in interpersonal circumstances than to do the reverse (donate nothing but take time out to help old ladies across the road, etc.). But any system for arriving at the fairer compromise just on the basis of those ordinal preferences over decisions could be manipulated into deciding differently just by introducing [39.9% and egoistic] or [0.1% and selfless] as a bill, or whatever. The cardinal aspect of the total utilitarian's preference is key to being able to consistently decide what tradeoffs that philosophy would be willing to make.

(NB: I'm aware that I'm being terribly unfair to the object-level moral philosophies of egoism and total utilitarianism, but I hope that can be forgiven along with my terrible notation in service of the broader point)

Edit: gjm puts it better