My whole argument rests on a weaker reed than I first appreciated, because the definition of mutual information I linked is for univariate random variables. When I searched for a definition of mutual information for stochastic processes, all I could really find was various people writing that it was a generalization of mutual information for random variables in "the natural way". But the point you bring up is actually a step in the direction of a stronger argument, not a weaker one. Sampling the function to get a time series makes a vector-valued random variable out of a stochastic process, and numerical differentiation on that random vector is still deterministic. My argument then follows from the definition of multivariate mutual information.