Laser-plasma accelerators are prominent candidates for driving next-generation compact light sources, promising high-brightness, few-femtosecond x-ray pulses intrinsically synchronized to an optical laser, and thus are ideally suited for pump-probe experiments with femtosecond resolution. So far, the large spectral width of laser-plasma-driven beams has been preventing a successful free-electron laser (FEL) demonstration using such sources. In this paper, we study the application of an optimized undulator design and bunch decompression to large-energy-spread beams in order to permit FEL amplification. Numerically, we show a proof-of-principle scenario to demonstrate FEL gain in the vacuum ultraviolet range with electron beams from laser-plasma accelerators as currently available in experiments.