We present a quantum optics formalism to study the intensity power broadening of a semiconductor quantum dot interacting with an acoustic-phonon bath and a high-Q microcavity. Power broadening is investigated using a time-convolutionless master equation in the polaron frame, which allows for a nonperturbative treatment of the interaction of the quantum dot with the phonon reservoir. We calculate the full non-Lorentzian photoluminescence (PL) line shapes and numerically extract the intensity linewidths of the quantum-dot exciton and the cavity mode as a function of the pump rate and temperature. For increasing field strengths, multiphonon and multiphoton effects are found to be important, even for phonon-bath temperatures as low as 4 K. We show that the interaction of the quantum dot with the phonon reservoir introduces pronounced features in the power-broadened PL line shape, enabling one to observe clear signatures of electron-phonon scattering. The PL line shapes from cavity pumping and exciton pumping are found to be distinctly different, primarily since the latter is excited through the exciton-phonon reservoir. To help explain the underlying physics of phonon scattering on the power-broadened line shape, an effective phonon Lindblad master equation derived from the full time-convolutionless master equation is introduced; we identify and calculate distinct Lindblad scattering contributions from electron-phonon interactions, including effects such as excitation-induced dephasing, incoherent exciton excitation, and exciton-cavity feeding. Our effective phonon master equation is shown to reproduce the full PL intensity and the phonon-coupling effects very well, suggesting that its general Lindblad form may find widespread use in semiconductor cavity-QED.