Sony firmware updates can be flashed without root using
FlashTool; the "customised US" firmware for the C6616 can be downloaded from
this XDA post, and is claimed to work on the T-Mobile C6606.
The usual warranty T&C preclude using "modified/reengineered software", so although this is an official Sony update the fact that it was intended for a different XZ customisation would probably give T-Mobile a handy get-out-clause if a warranty claim was made.