To better know forces and deflections, you need to model this with FEA with soil springs and rock springs, based on lateral modulus. The caisson embedment should be at least the minimum bedrock penetration prescribed be the soils report.
The soil/rock lateral stiffness is materially non-linear (P-y curve) but for the small caisson head deflection most SE's are going to tolerate, a linearly elastic soil/rock spring is OK in my opinion. Doing material non-linear FEA based on a geotechnical parameters known to one or two significant figures is "putting lipstick on a pig".
Consider whether the TOP of your caisson is "fixed head" or "free head" and I usually model the caisson as having a cracked cross section (Ieff = 0.5 Igross) as a simplification.
The soil stiffness immediately at the top of the pier, the depth of overburden and the pier head BC's affect forces and deflections for laterally loaded piers MUCH more so than the bedrock penetration is my experience.