She did the right thing rescuing the Emperor. Starfleet should be about saving lives and reducing harm. Her later justifications about maybe getting intel on tactics is also valid.
In Burnham's own universe, perhaps. But in this case, Georgiou had accepted her fate in her own mirror universe. Bringing her back to the P.U. because Burnham was sad she lost her mommy-figure-mentor was really stupid. Georgiou had already proven her utter ruthlessness; did Burnham think of the consequences of what she was doing? Hey, I'm a gonna save and bring over a mini-Hitler-type to an unprepared Starfleet. Gah.
Burnham just wanted to save Georgiou THIS TIME, even though it was a totally different Georgiou. Where's the fucking logic in that? Sarek must be smacking himself upside the head; Burnham may be book smart but her logic leaves much to be desired.