Azar, to be fair, only possessed the most basic qualities necessary for a spy. Although he managed to stand out in the vast abyss of the Shah Intelligence Bureau, in Anton's eyes, this guy was just a hair's breadth away from being an idiot.
Joking aside, as a world-stirring organization, their leader was captured; how could they not desperately keep secrets and block information? How could they let Azar and Anton relay the news?
Even if Baghdadi ultimately died, that would be a matter to discuss after his death, not to mention that Baghdadi might not even be dead yet.
That's why Anton wouldn't waste words, to avoid the embarrassment of refusal from the Viper, which was secondary; what mattered was provoking the Viper, who might actually lock them up, which would be troublesome, even more so if they were silenced forever.