We study the scaling limit of the rank-one truncation of various beta ensemble generalizations of classical unitary/orthogonal random matrices: the circular beta ensemble, the real orthogonal beta ensemble, and the circular Jacobi beta ensemble. We derive the scaling limit of the normalized characteristic polynomials and the point process limit of the eigenvalues near the point 1. We also treat multiplicative rank one perturbations of our models. Our approach relies on a representation of truncated beta ensembles given by Killip and Kozhan, together with the random operator framework developed by Valkó and Virág to study scaling limits of beta ensembles.