We present the latest upgrades of our cryogenic multi-Penning-trap stack to implement quantum logic cooling and detection techniques for $g$-factor measurements of (anti-)protons within the BASE collaboration. This updated trap system has been designed to cool single protons to their motional ground state, a key feature for the implementation of quantum logic spectroscopy on (anti-)protons in Penning traps. For this purpose, a double-well potential will be generated in our trap system in order to implement free Coulomb coupling between a single proton and a $^9$Be$^+$ ion, allowing sympathetic cooling of single protons, as well as a spin-motion SWAP gate for spin state detection. The application of these techniques to antiprotons later will allow to increase the sampling rates of current (anti-)proton $g$-factor measurements, and thus contribute to the search for CPT violating coefficients acting on baryons in the Standard Model extension.