Thomson scattering (TS) from electron beams produced in laser-plasma accelerators may generate femtosecond pulses of quasi-monochromatic, multi-MeV photons. Scaling laws suggest that reaching the necessary GeVelectron energy, with a percent-scale energy spread and five-dimensional brightness over 10^16 A/m^2, requires acceleration in centimeter-length, tenuous plasmas (n ~ 10^17 cm^-3), with petawatt-class lasers. Ultrahigh per-pulse power mandates single-shot operation, frustrating applications dependent on dosage. To generate high-quality near-GeV beams at a manageable average power (thus affording kHz repetition rate), we propose acceleration in a cavity of electron density, driven with an incoherent stack of sub-Joule laser pulses through a millimeter-length, dense plasma (n ~ 10^19 cm^-3). Blue-shifting one stack component by a considerable fraction of the carrier frequency compensates for the frequency red shift imparted by the wake. This avoids catastrophic self-compression of the optical driver and suppresses expansion of the accelerating cavity,

avoiding accumulation of a massive low-energy background. In addition, the energy gain doubles compared to the predictions of scaling laws. Head-on collision of the resulting ultrabright beamswith another optical pulse produces, via TS, gigawatt gamma-ray pulses having a sub-20% bandwidth, over 10^6 photons in a microsteradian observation cone, and the observation cone, and the mean energy tunable up to 16 MeV.