In this work, we combine an isogeometric Petrov-Galerkin approach [1] that employs approximate dual spline functions [2] with the Hellinger-Reissner principle to eliminate the effect of membrane locking for Kirchhoff-Love shells. We consider the displacement and strains as independent variables and interpolate the latter with a lower-order spline basis to overcome locking. We discretize all test functions of the strain variables with an approximate dual spline basis that is smooth, has local support, and satisfies approximate bi-orthogonality in the mapped domain with respect to a trial space of B-splines. Lumping the projection matrix on the left-hand side of the strain projection equations yields an identity matrix, eliminating the need for matrix inversion, without compromising higher-order accuracy of the projection. We demonstrate the performance of our approach in terms of spatial accuracy via convergence studies and spectral analysis of curved beam and shell benchmarks.