Question: Did the argument on whether or not to make the bombers' photos public really take place?
BaconIsMyBFF
6th Jan 2018
Patriots Day (2016)
6th Dec 2017
Patriots Day (2016)
Question: Were the bombers members of a terrorist organization?
Answer: They were not members of any terrorist group. They taught themselves how to build bombs with internet research and acted alone.
Answer: In short, no it likely did not. In real life the FBI voluntarily released the photos of the suspects to the press with a message to the public to help identify the suspects. The FBI statement also claimed they were doing this to help protect people wrongly identified as suspects on social media. While it is possible there could have been some back and forth privately regarding the photo release, it is also relatively common for law enforcement to release photos of suspects to the public for the same reasons as were done in the Boston Marathon bombings.
BaconIsMyBFF
According to history vs. Hollywood there really was a debate on whether or not to release the photos.