Even though the Red Angel turns out not to be Burnham, the ed Angel still possesses technology far beyond Starfleet's capabilities at the time. Its sensors would have been able to detect a support team in the next room from where Burnham was allegedly "dying," plus two Starfleet vessels in orbit, and been able to deduce the trap and not respond to the bait.
If the Red Angel is, as the crew suspects, Commander Burnham, then the Angel would have knowledge that the trap they set for her was not authentically putting Burnham's life in danger and she would not respond to the supposed "threat" as though it were real.