Is Don Juan in Byron's long poem Don Juan a Byronic Hero?

Don Juan does not strike me as a Byronic hero.
The Byronic hero has qualities that reflected Lord Byron himself. As outlined in Byron's The Corsair, the Byronic hero is "that man of loneliness and mystery" who is "scarce seen to smile, and seldom heard to sigh." The Byronic hero is closed off to the world, shrouded in questions, and capable of a dangerous edge.
Such worldly traits do not describe Don Juan. One reason is because he is quite naive. If the Byronic hero is the puppeteer who makes sure that everyone else dances to his tune, Don Juan is a puppet. His trusting nature makes him easily malleable in the hands of other people. Don Juan lives up to his namesake as "the true one" when he falls in love with Donna Julia. Later on, Don Juan shares the most innocent of loves with Haidee, the pirate's daughter. Falling in love requires a selfless submission towards another person. A true Byronic hero would not be so trusting. Such an innocent and pure embrace of love is not in the makeup of the Byronic hero. The Byronic hero is the type who "loves 'em and leaves 'em." There is an edge to the Byronic hero that makes loving him impossible.
The Byronic hero does not easily forge connections with other human beings. He stands on his own and lives life on his own terms.  These terms do not include absorbing emotions from other people. However, Don Juan is susceptible to having emotional connections imprinted on him. For example, when Don Juan is in the slave market, the depth of Haidee's love has imprinted itself on him. This is seen in the way he talks with John Johnson about how much they mean to one another. When Don Juan is purchased by Gulbeyaz, he is expected to please her. However, Don Juan is loyal to Haidee and is direct about the extent of his connection to her: "The prisoned eagle will not pair, nor I/ Serve a sultana's sensual phantasy." These would not be the words of the Byronic hero, who has spurned people's attempts to enter his heart. Don Juan does not take the Sultana to be his lover because his heart already belongs to another. This level of emotional connection is not readily found in the Byronic hero.