He is connecting to a Car's computer. When he says computer, I assumed he meant that one. So he wouldn't have any conflict with the 'computer' if he only connects the TX to the Arduino RX. This would also allow him to connect only his TX to the Display with the possible limitations I mentioned above.
I found that you only have disconnect the Arduino RX line to upload a sketch when using these displays because you have to hold the reset pin high. As a result, the display is off and any data sent back to the IDE via the TX (is there any?) during upload would not interfere with the Display.