After a little while, my thoughts for a layperson are a bit clearer, and I thought I'd re-iterate them here.
These are different uncertainties to deal with. We must be clear: where we are pretty sure about the model spec, Bayesian inference is about quantifying the uncertainty in the parameter values. Under this paradigm, if we use more data, we get narrower posterior distributions, and if we use less data, we get wider posterior distributions. If we split the data, we're just feeding in fewer data points to the model; if we don't, then we're just feeding in more data points.