We present the results of high-pressure transport measurements on the anion-mixed molecular conductors (DIETSe)${}_{2}M{\mathrm{Br}}_{2}{\mathrm{Cl}}_{2}$ [DIETSe = diiodo(ethylenedithio)tetraselenafulvalene; $M=\mathrm{Fe}$, Ga]. They undergo a metal-insulator (M-I) transition below 9 K at ambient pressure, which is suppressed by applying pressure, indicating a spin-density-wave (SDW) transition caused by a nesting instability of the quasi-one-dimensional (Q1D) Fermi surface, as observed in the parent compounds (DIETSe)${}_{2}M{\mathrm{Cl}}_{4}$ ($M=\mathrm{Fe}$, Ga). In the metallic state, the existence of the Q1D Fermi surface is confirmed by observing the Lebed resonance. The critical pressures of the SDW, ${P}_{\mathrm{c}}$, of the $M{\mathrm{Br}}_{2}{\mathrm{Cl}}_{2}$ ($M=\mathrm{Fe}$, Ga) salts are significantly lower than those of the the $M{\mathrm{Cl}}_{4}$ ($M$ = Fe, Ga) salts, suggesting chemical pressure effects. Above ${P}_{\mathrm{c}}$, field-induced SDW transitions appear, as evidenced by kink structures in the magnetoresistance (MR) in both salts. The ${\mathrm{FeBr}}_{2}{\mathrm{Cl}}_{2}$ salt also shows antiferromagnetic (AF) ordering of $d$ spins at 4 K, below which significant spin-charge coupling is observed. A large positive MR change up to 150% appears above the spin-flop field at high pressure. At low pressure, in particular below ${P}_{\mathrm{c}}$, a dip or kink structure appears in MR at the spin-flop field, which shows unconventionally large hysteresis at low temperature $(T<1$ K). The hysteresis region clearly decreases with increasing pressure towards ${P}_{\mathrm{c}}$, strongly indicating that the coexisting SDW plays an important role in the enhancement of magnetic hysteresis besides the random exchange interaction.