The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here, we present new scalable MFS formulations for the corresponding elastance and mobility problems. The elastance problem computes the potentials of conductors with given net charges, while the mobility problem—crucial to rheology and complex fluid applications—computes rigid body velocities given net forces and torques on the particles. The key idea is orthogonal projection of the net charge (or forces and torques) in a rectangular variant of a “completion flow.” The proposal is compatible with one-body preconditioning, resulting in well-conditioned square linear systems amenable to fast multipole accelerated iterative solution, thus a cost linear in the particle number. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. Our several numerical tests include a suspension of 10,000 nearby ellipsoids, using 2.6 x 10⁷ total preconditioned degrees of freedom, where GMRES converges to five digits of accuracy in under two hours on one workstation.