No, the correct pronoun is 'him.' Chelsea Manning, despite his self-perception and despite his name change, is and always will be a man. Gender is an objective, physical, biological fact, not a subjective experience.
There are people who believe that they are disfigured even though they are perfectly normal looking; they are not disfigured, even though they believe so. There are people who believe that their limbs are not their own; those limbs are theirs, whatever their perception. There are people who believe that they are the Messiah; they are not.
When talking to someone, it may be therapeutic (or simply kind) to use the words he prefers even if they are incorrect, but when talking about someone, there is no need to humour his error.