The "geometrical approach" of splitting up the chain is actually well suited for my use case. I'm using this arm as a fancy forklift for loading train cars in a circle. This means what I'm essentially dealing with is a three-chain system.
1. The base turntable at the center of the track circle rotates the rest of the arm to point at the target (train car, a stack of shelves, etc).
2. The first two joints are a two-bone IK chain in two dimensions, bringing the base of the wrist near the target.
3. The wrist (Akiyuki wrist) then has the job of "canceling out" the angle at which it attaches to the arm, in order to continuously keep the forklift end level.
Analyzing this, the first 1DOF chain is very simple, and does not need any kinematic algorithm beyond a P loop. The second is slightly tricky, but 2-dimension 2DOF solving basically comes down to constructing a triangle given three sides, which basic trigonometry can easily handle. The third chain, however, will present more difficulties, at least if the wrist rolls. If it stays in two dimensions it's simply an arithmetic calculation based on the angle of the "forearm".