The Remote Transceiver Module (RTM) is located in the passenger side of the headliner next to the Center High Mount Stop Lamp (CHMSL). The RTM receives the Remote Keyless Entry (RKE) data from the key fob and transmits it to the BCM for functions such as lock/unlock, panic, tailgate release, etc. The RTM also interprets the high frequency signal from the key fob to allow for the passive start to function.
From your description, the RTM or it's connection was damaged during the installation, or your coat hanger contacted the back side of the connector and opened a fuse. Start by checking Fuse F32 in the BCM. This fuse provides power to the RTM and APIM (Sync Module). If the fuse is good, lower the headliner and inspect the RTM and it's harness and connection.
The BCM is #1 and the RTM is #2 in the illustration:
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