This work examines the effect of dark matter haloes on the fate of bar-driven escaping patterns in disc galaxies. In this study, a three-dimensional gravitational model with a strong bar profile has been considered and examined separately for the following dark halo profiles: NFW and oblate. These profiles are chosen to model massive and low-mass disc galaxies, respectively. In both cases, a bar-driven escape mechanism has been identified near the saddle points of the phase space, which correspond to the bar ends. This bar-driven escaping motion has been analysed via orbital and Poincaré surface section maps. Moreover, with a choice of initial condition in the vicinity of escape saddles, the variation of its maximal Lyapunov exponent value against the dark halo parameters such as mass, size, circular velocity and nature has been studied. This helps to determine the sensitivity of the bar-driven escaping patterns to the dark halo parameters. Our results state that NFW dark haloes support the formation of bar-driven grand design spiral arms (as we have seen in massive disc galaxies) only if there is an excess energy generation source other than the baryonic feedback generated due to the accretion of the central supermassive black hole. In this regard, active galaxies might be one of the potential contenders where NFW profiles may be a better fit for modelling dark haloes. On the other hand, oblate dark haloes support the formation of bar-driven less prominent spiral arms in low-mass disc galaxies like S0, ultra-compact dwarfs, etc.