This will not be a good answer, but hopefully an almost decent breadcrumb. What you're describing has roots in robotic motion planning, which extends the basic notion of pathfinding to include volumetric concerns and degrees of freedom. A useful term to get familiar with is "configuration space", which offers some reduction of complexity. I recommend using "motion planning" as your search term, and maybe start here: https://en.wikipedia.org/wiki/Motion_planning

You're onto a tough problem but it'll be fun. Good luck!