Earlier quoted context omitted.
Robin Hanson has a paper where he argues that such attempts at manipulation won't work: https://mason.gmu.edu/~rhanson/biashelp.pdf Based on the abstract, I think the essential argument is that a market manipulator acts as "dumb money", attracting "smart money" who are happy to profit by bringing the market back in line with reality.
I think what I got confused by is the idea that a proposal that is “popular with the market” is popular because it reduces inequality and not because it has good financial returns. Take the inequality example that you mention. Reducing inequality might require large government spending, like free higher education, or extensive government-provided healthcare. This requires funding which pushes up interest rates. Someo…
One approach would be to have multiple markets, that track endpoints like inequality and interest rates separately for any given candidate policy.
In the best case, prediction markets could give you accurate forecasts on a large number of candidate policies, and you could select a policy that the market thinks does a pretty good job according to lots of relevant endpoints. (My hunch is that our current system is bad at searching for such win/win policies, preferring instead the drama that comes from clashing over win/lose policies. For example, you mention free higher education, but how much does the average college grad really use or remember from their degree? I'd argue there is a lot of territory in apprenticeships that is currently unexplored, and smart policy could facilitate this -- especially if we start with small-scale experimentation.)
You're absolutely correct that the market can't help with moral questions such as how much government spending is justifiable in pursuit of reduced inequality. Robin Hanson proposes eventually moving to a system where the job of elected officials is to manage metrics related to what voters want out of government policies, and markets decide the rest. In slogan form: "Vote on values, bet on beliefs." https://mason.gmu.edu/~rhanson/futarchy.html