The In-Gas-jet Laser Ionization and Spectroscopy (IGLIS) technique relies on narrow-bandwidth, high-peak-power, short-pulse-length (≈10 ns), and high-repetition-rate laser pulses to probe, precisely and efficiently, the hyperfine structure of medium-heavy and heavy isotopes, embedded in a supersonic jet. The power and repetition rate requirements of the laser system are met by combining ≈100 W, 8 ns pulse width, 10 kHz commercial Nd:YAG pump lasers with a single-mode continuous wave seeded Pulsed Dye Amplifier (PDA). The common multi-longitudinal-mode operation of these Nd:YAG pump lasers causes, however, undesirable frequency sidebands in the output spectrum of the PDA system, hindering the attainable spectral resolution, a correct interpretation, and an accurate analysis of the hyperfine spectra. In this article, a new prototype Nd:YAG laser is presented, which combined with the PDA system is capable of providing quasi-transform-limited laser pulses at 10 kHz, with only limited losses in laser power. This system reduces any spectral sideband amplitude below a proven upper limit of 0.2% with one order of magnitude extra reduction expected based on simulations. A full characterization of both the Nd:YAG and PDA laser systems is done by studying the temporal and frequency behavior in detail. This study is finalized by a performance benchmark of this combined laser system in the hyperfine spectroscopy of copper isotopes, showcasing its applicability for future IGLIS studies.