  Photon engineering can be exploited to control the nonlinear evolution of the
drive pulse in a laser-plasma accelerator (LPA), offering new avenues to tailor
electron beam phase space on a femtosecond time scale. One promising option is
to drive an LPA with an incoherent stack of two sub-Joule, multi-TW pulses of
different colors. Slow self-compression of the bi-color optical driver delays
electron dephasing, boosting electron beam energy without accumulation of a
massive low-energy tail. The modest energy of the stack affords kHz-scale
repetition rate at manageable laser average power. Propagating the stack in a
preformed plasma channel induces periodic self-focusing in the trailing pulse,
causing oscillations in the size of accelerating bucket. The resulting periodic
injection generates, over a mm-scale distance, a train of GeV-scale electron
bunches with 5D brightness exceeding $10^{17}$ A/m$^2$. This unconventional
comb-like beam, with femtosecond synchronization and controllable energy
spacing of components, emits, via Thomson scattering, a train of highly
collimated gigawatt $\gamma$-ray pulses. Each pulse, corresponding to a
distinct energy band between 2.5 and 25 MeV, contains over $10^6$ photons.
