You could probably let the schedule of the perturbations be a sequence ε→0, such that ∑ε = ∞, and such that ∑ηε < ∞ with probability 1, where η is IID, bernoull

You could probably let the schedule of the perturbations be a sequence ε→0, such that ∑ε = ∞, and such that ∑ηε < ∞ with probability 1, where η is IID, bernoulli on {-1, 1} (e.g. a square-summable sequence will do). I suspect this will be a fairly good start. Note that, then it should in general be independent of the chosen parameter, since the added noise converges in probability to a constant whenever the distribution is close to uniform, but can sum to any real value with consistent bias.