I haven't been able to find any official commentary on the decision to draw attention to Bail during the scene, either from director Gareth Edwards, or actor Jimmy Smits. The closest I've found is just Jimmy Smits confirming that he liked his scenes and didn't do any alternate takes. However, here are some possible reasons:
His appearance is likely intended as fanservice.
He's the first character only seen in the prequels to appear in the film, so his appearance in the scene could be intended to draw a connection between the prequels and original trilogy. Bail appears in Attack of the Clones and Revenge of the Sith, and is mentioned in A New Hope in Leia's iconic message to Obi-Wan Kenobi. In the words of RedLetterMedia, "I clapped! I clapped when I saw [Bail Organa]!"
Why was the pacing kind of clumsy, if it's fanservice? It's unlikely this will ever be officially addressed, but it's worth noting that this scene as described in the official novelization, released at the same time as the film, doesn't include Bail at all. His appearance in this scene could have been a last-minute addition in editing.
The scene is establishing that Bail is involved with the Death Star mission.
The music that plays during his first appearance in Rogue One is the Force Theme, with a very similar orchestration to the version that plays during Obi-Wan's first appearance in A New Hope, indicating that the scene is meant to remind the audience that he's involved with the plot to get the Death Star Plans. As mentioned, his sole previous mention in the context of A New Hope is that he wanted to get the Death Star plans to Obi-Wan Kenobi, so the scene is establishing that 1) this is an important character you're meant to take note of, and 2) you've previously heard of him in the context of the Death Star plans.
He may be sympathetic to Jyn.
This is more speculative, since he doesn't actually interact with Jyn during the film, other than their eye contact during this scene, but in this scene, the Rebel command is interrogating Jyn about her relationship with her father, an Imperial collaborator. Since Bail's own adopted daughter, Leia, is about Jyn's age and also (secretly) has a father who's a high-ranking member of the Empire, the focus on his appearance may be meant to imply that he has some sympathy for Jyn, even though he's part of Alliance Command. (source)