> In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears.
You are moving the bar and no longer talking about classical mechanics.
We know classical mechanics is not the correct theory, the point is that one of the ways it is incorrect is that it allows for mathematical singularities in its solutions. If you start slapping restrictions on the theory: "finite force propagation, energy and speed limits, regularity of solutions, continuity of fields, etc...." to prevent those singularities from appearing then it is no longer classical mechanics. It is something else.
> If we know all the details about the system then we can predict its behavior with arbitrary precision. That knowledge, of course, must include all the infinitely fast objects.
Again you aren't talking about Classical Mechanics. Your Demon wants to predict what will happen to the ball at time t_2, and wants to make that prediction based on the state of the universe at time t_0. Suppose that the correct prediction is "the ball does not exist at time t_2, because an Invader appears at time t_1 \in (t_0, t_2) and vaporizes the ball before t_2." If your Demon is capable of making such a prediction then he/she must be able to:
Express in the notation of classical Newtonian mechanics the position and velocity of the Space Invader at time t_0, so that I can deterministically show that the invader MUST begin to slow at time t_1 and destroy the ball prior to time t_2.
You CANNOT do so because v=infinity and x=everywhere (or x=emptyset) is not a valid expression of position and velocity of a body in Newtonian Mechanics. In Newton's formulas, the infinitely fast object DOES NOT EXIST within the classical universe, but his formulas allow it to be the finite-time limit of a classical process.