In a perfect world, I would enter the active soil parameters from the geotech report (edge moisture distance, heave, etc.) and the program would perform the calcs, similar to a spreadsheet or design program based on the PTI procedure. I ran several STAAD Foundation Advanced models considering the loading approaches from the original post above, and the center lift case seemed to run accurately. With the soil springs applied to an inset control region, the moment diagrams followed the profile shown in the USACE ribmat design criteria for center lift, and no errors were reported in the analysis. As expected, the soil bearing pressures were highest at the corners where the load cantilevers in both directions but tapered back quickly.
For edge lift, I had a lengthy conversation with a geotech before developing the model. While this might appear to be a simple load condition, it is far from it. The swell pressure is not uniform, but is tied to the ability of the soil to expand with changes in moisture content. A stiff foundation would confine or "push back" against the soil, limiting its expansion and resulting in high exerted soil pressure. A flexible foundation would bend upward, allowing the soil to reach its full volume-change potential which results in low swell pressure. So the swell pressure and foundation stiffness are related (deflection inversely related to swell pressure), and this is not captured in the simplified loading described in the OP. It represents a theoretical "worst case" assuming the soil expansion is limited to a value resulting in maximum swell pressure, but does not capture the real-world behavior or bearing pressure. The model just sees it as an uplift load that is trying to pick the mat up, instead of the soil pressure increasing below the foundation. Therefore the service load bearing pressure and the edge deflections are fictitious numbers. The model value is limited to providing worst-case design values for concrete reinforcement.
It's not a perfect solution, so I'm still interested to know if there's a better way to approach the problem.