Do batteries have a delay when adjusting to a new load?
Yes, there definitely are time constants associated with establishing new equilibrium reactant distributions when the load on a battery changes. The density of the reactants directly affects the internal impedance of the battery, and therefore its terminal voltage. These effects get larger as the battery ages and the overall density of reactants drops.

However, manufacturers don't spend much time characterizing this, so you're going to have to make your own measurements and adjust your test procedure to suit.