SHORT UPDATE: After doing some tests with the GPS module on a breadboard, we are designing an extension board. Everything needed for running the module is on the board. Especially the LDO, because of the manufacturer's requirement of a high PSRR. 

The manufacturer recommends: "An LDO with a high PSRR should be chosen for good performance. In addition, a TVS diode, and a combination of a 4.7 μF, 100 nF and a 33 pF decoupling capacitor network should be added near the VCC pin. The lowest value capacitor should be the closest to module pins."