Intense sub-cycle electromagnetic pulses allow one to drive nonlinear processes in matter with unprecedented levels of control. However, it remains challenging to scale such sources in the relativistic regime. Recently, a scheme that utilizes laser-driven wakes in plasmas to amplify and compress seed laser pulses to produce tunable, carrier-envelope-phase stable, relativistic sub-cycle pulses has been proposed. Here, we present parametric studies of this process using particle-in-cell simulations, showing its robustness over a wide range of experimentally accessible laser-plasma interaction parameters, spanning more than two orders of magnitude of background plasma density. The method is shown to work with different gas-jet profiles, including structured density profiles and is robust over a relatively wide range of driver laser intensities. Our study shows that sub-cycle pulses of up to 10mJ of energy can be produced.