Of these four characters, Dr. Stockman is the only one I would argue does not have the "Romantic hero" qualities you allude to, as the others do.
Ibsen's works date from the period after the Romantic movement had basically ended. Though this in itself doesn't disqualify Stockman from being a Romantic hero, if we look at the qualities of the others you've named in some detail, their differences from Ibsen's character should become apparent.
Byron's heroes such as Childe, Harold, and Manfred are men who are outcasts, either inherently so or because they have imposed a kind of exile upon themselves out of a sense of regret and of not belonging in conventional society. Harold sees himself a "sinner," a transgressor. "And none did love him," we are told. He takes off on a journey through Europe as if hoping to find some key to the meaning of life that his empty, riotous existence in England has not afforded him. Manfred, similarly, is self-banished to a remote place in the Alps, also trying to capture an ultimate experience, in his case, paradoxically, "forgetfulness." In both cases these men strive for an impossible, unreachable quality or state beyond normal human existence, as Goethe's Faust does. The attempt to gain access to the "infinite" is typical of the Romantic mindset. Mary Shelley's Viktor Frankenstein fits in perfectly with this trend because he dares to create life, not merely to transcend ordinary experience in his own life. All of these characters are "unrealistic" in their goals, and are failures in some sense, representing the pessimistic side of the Romantic movement in spite of their idealism.
In Ibsen's Enemy of the People, Dr. Stockman is "unrealistic" as well in not anticipating or caring about the extent to which others will oppose him when he attempts to reveal the health dangers of the town's baths. He's therefore an "idealist," but significantly not in the manner of the earlier Romantic heroes. His thinking is grounded in reality, and a prosaic reality at that. He simply wants to reveal a scientific "truth" in his pure, incorruptible way, without understanding the practical consequences of doing so. It's a different matter entirely from the abstract striving for the "impossible" we see in the genuinely Romantic hero. The realistic science upon which his conclusions are founded is in some sense the opposite of the Gothic science of Frankenstein and other works of the Romantic period such as those of E.T.A. Hoffmann.
Still, the common element between Stockman and the Byron and Shelley characters is their Otherness. They are all outcasts, rejected by the rest of mankind, pursuing courses of their own. Admittedly, then, Stockman's aloneness does connect him with Romantic heroism, though by his time, the late nineteenth century, man had entered a materialistic, technological age that negated the mysteries the Romantic writers based their aesthetic on.