  Strong coupling of semiconductor spin qubits to superconducting microwave
resonators was recently demonstrated. These breakthroughs pave the way for
quantum information processing that combines the long coherence times of
solid-state spin qubits with the long-distance connectivity, fast control, and
fast high-fidelity quantum-non-demolition readout of existing superconducting
qubit implementations. Here, we theoretically analyze and optimize the
dispersive readout of a single spin in a semiconductor double quantum dot (DQD)
coupled to a microwave resonator via its electric dipole moment. The strong
spin-photon coupling arises from the motion of the electron spin in a local
magnetic field gradient. We calculate the signal-to-noise ratio (SNR) of the
readout accounting for both Purcell spin relaxation and spin relaxation arising
from intrinsic electric noise within the semiconductor. We express the maximum
achievable SNR in terms of the cooperativity associated with these two
dissipation processes. We find that while the cooperativity increases with the
strength of the dipole coupling between the DQD and the resonator, it does not
depend on the strength of the magnetic field gradient. We then optimize the SNR
as a function of experimentally tunable DQD parameters. We identify wide
regions of parameter space where the unwanted backaction of the resonator
photons on the qubit is small. Moreover, we find that the coupling of the
resonator to other DQD transitions can enhance the SNR by at least a factor of
two, a ``straddling'' effect that occurs only at nonzero energy detuning of the
DQD double-well potential. We estimate that with current technology,
single-shot readout fidelities in the range $82-95\%$ can be achieved within a
few $\mu\textrm{s}$ of readout time without requiring the use of Purcell
filters.
