We present the second prototype of a time-to-digital (TDC) ASIC for the upgrade of the ATLAS Monitored Drift Tube (MDT) detector for High-Luminosity LHC operations. Compared to the first prototype, triple modular redundancy has been implemented for the configuration and flow control logic. The total power consumption is increased by less than 10 mW while achieving the same time resolution and channel uniformity. A mini-DAQ system has been built to verify the front-end electronics chain with the new prototype together with other ASICs and boards in triggered mode. Cosmic ray tests with a small-diameter MDT chamber indicate that the configuration and data transmission of the readout electronics perform well. It is expected that this prototype design will be used in the final production.