We consider the action of invariant differential operators on the symmetric algebra of a mutiplicity-free representation of a basic classical Lie superalgebra. We show that for the general class of examples that arise from the TKK construction (where the representation space is indeed a Jordan superalgebra), there is a distinguished basis of the algebra of invariant differential operators (known as the Capelli basis) whose spectrum is given by suitable specialisations of super analogues of Macdonald polynomials defined by Sergeev and Veselov.