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 pre-formed 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.