  Both quantum transport measurements in the Pauli blockade regime and
microwave cavity transmission measurements are important tools for spin-qubit
readout and characterization. Based on a generalized input-output theory we
derive a theoretical framework to investigate how a double quantum dot (DQD) in
a transport setup interacts with a coupled microwave resonator while the
current through the DQD is rectified by Pauli blockade. We show that the output
field of the resonator can be used to infer the leakage current and thus obtain
insight into the blockade mechanisms. In the case of a silicon DQD, we show how
the valley quasi-degeneracy can impose limitations on this scheme. We also
demonstrate that a large number of unknown DQD parameters including (but not
limited to) the valley splitting can be estimated from the resonator response
simultaneous to a transport experiment, providing more detailed knowledge about
the microscopic environment of the DQD. Furthermore, we describe and quantify a
back-action of the resonator photons on the steady state leakage current.
