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Formal and experimental studies of Fabry-Perot cavities operating in burst mode are conducted in the context of Compton scattering. A dedicated optimization procedure for their use in this context is described for the first time, to the best of our knowledge. Provided that the number of bunches in the electron-beam train and their time separation are known, the parameters related to the optical system are chosen to maximize the ratio of the total intracavity laser-beam energy used for collisions to the total input laser beam seeded into the cavity operated in burst mode. A good qualitative agreement is obtained between the model and the experimental data. It is shown by studying the upgrade of an existing machine, STAR. Here a new optical system with increased performance without a large increase in complexity is obtained. This result paves the way towards future designs of high-flux, high-quality, low cost and compact x-ray facilities based on Compton scattering.