The Stokes flow occurring within a non-neutrally buoyant spherical droplet translating by buoyancy through an immiscible liquid which is undergoing simple shear is shown to be chaotic under many circumstances for which the droplet translates by buoyancy through the entraining fluid. This flow is easily produced, for example, when the droplet rises (or falls) through the annular space of a vertical concentric-cylinder Couette viscometer or through a vertical Poiseuille flow. The parameters studied include: (i) droplet/bulk fluid viscosity ratio; (ii) shear strength/bubble rise velocity ratio; and (iii) the angle between the translational bubble velocity vector and the vorticity vector characterizing the undisturbed shear. Streamlines existing within a droplet that translates perpendicular to this vorticity vector are shown to be non-chaotic for all choices of physical parameters. Other relative orientations frequently contain chaotic trajectories. When solute initially dissolved within the droplet is extracted into the bulk fluid, the resulting overall mass-transfer coefficient (calculated via generalized Taylor dispersion theory) quantifying the extraction rate at asymptotically long times is shown to be significantly higher in the chaotic flow case.