"Henry Charles Bukowski Jr's nom de plume was Charles Bukowski, and Charles Bukowski wrote a form of imaginative, richly embellished, reportage about his own noir existence. Hank Chinaski, an iconoclast poet who feverishly abused alcohol with a fierce joie de vivre, appears as Bukowski's avatar in his skid-row prose and poetry. Chinaski seems to be a thinly veiled Bukowski, an outwardly uncouth, gruff and pedestrian writer; however, Charles Bukowski was the persona of Henry Charles Bukowski Jr. Chinaski was the persona of a persona." (Michael Basinski, from the preface of 'Charles Bukowski, the King of the Underground' by Abel Debritto).
With the above in mind, I tentatively suggest that he signed his letters to John Martin with Henry because it was the man himself (Henry Charles Bukowski Jr) writing to his publisher/editor and not the fictionalised character of Charles Bukowski that he had invented for literary purposes.