Surely it isn't a requirement of instrumentalism that the measurements we make must only be one-dimensional.

Yeah, that would be a dangling requirement.

More specifically, your claim that the ultimate measurement results must be integers doesn't rule out the possibility that each measurement result gives a pair of integers (instead of a dial moving along a single dimension, think of one that can move in two dimensions).

That would be an interesting counterfactual universe, where you need at least two numbers to make sense of anything. I cannot quite imagine how this would work, but maybe this is a limit of my imagination. In the Universe we are currently stuck with, with the classical world described by classical mechanics based on real numbers, and the quantum world manifesting itself only through classical measurements, any useful model must ultimately lead to a set of readings off some dial, each of which should make sense separately.