Anti-parity-time-symmetric Hamiltonians show an enhanced sensitivity to external perturbations that can be used for high-performance angular velocity sensing. Dissipative coupling is a valuable way for realizing anti-PT-symmetric Hamiltonians with optical resonators and is usually obtained by means of auxiliary waveguides. Here, we model and experimentally show the dissipative coupling between two counterpropagating modes of a single resonator, by means of a Bragg-grating placed in the feeding bus. The proposed solution enables the possibility of accurately designing the dissipative coupling strength in integrated non-Hermitian gyroscopes, thus providing high flexibility in the design of the proposed sensor. Moreover, we theoretically and experimentally demonstrate that the dissipative coupling between two counterpropagating modes of the same resonant cavity can give rise to an asymmetric Fano resonance.