We demonstrate a Ho:YLF regenerative amplifier (RA) overcoming bifurcation instability and consequently achieving high extraction energies of 6.9 mJ at a repetition rate of 1 kHz with pulse-to-pulse fluctuations of 1.1%. Measurements of the output pulse energy, corroborated by numerical simulations, identify an operation point (OP) that allows high-energy pulse extraction at a minimum noise level. Complete suppression of the onset of bifurcation was achieved by gain saturation after each pumping cycle in the Ho:YLF crystal via lowering the repetition rate and cooling the crystal. Even for moderate cooling, a significant temperature dependence of the Ho:YLF RA performance was observed.