Well, there was the time that Karasuma couldn't choose between Tenma and instant curry and then told Harima to go after her for him because there was a sale on curry at a nearby restaurant... Karasuma loves curry as much or more than Tenma. That's at least three-quarters-assed if not half-assed.Even we, the audience, lack a full understanding of the situation. Is Tenma crying because Karasuma rejected her without a reason? Or is she crying because he told her about his condition?
There's a certain arrogance to Harima's actions, as Karasuma rightly points out. Harima's frustration comes from the fact that he believes that he loves Tenma more than Karasuma does. But how can Harima call Karasuma's feelings "half-assed" when he doesn't even understand the nature of Tenma and Karasuma's relationship to begin with?
More importantly, from Harima's perspective Karasuma is just jerking Tenma around. Karasuma told Harima that he would ask her to be his girlfriend after Harima urged him to seize love and manga, and now Karasuma has rejected Tenma? It's very bizarre behavior and even more bizarre to Harima who knows less than we do.
This isn't a problem for Harima. He gets along great with Karasuma when the issue of Tenma isn't between them. What bothers him is Karasuma's treatment of Tenma (choosing curry over her, mysteriously rejecting her, etc.)For example, there's a fairly important moment at the end of Ch. 119 where Karasuma explains that although he feels strong emotions on the inside, he is physically unable to show them. Tenma understands him in spite of this, which is why she is able to love him when so many others simply dismiss him as weird.
Not possessive? All romantic love is possessive to some degree. Karasuma didn't want Harima to see Tenma when she came over to his house. Karasuma also tried to run from Tenma's confession; he doesn't want to let her decide.And the very biggest revelation came in Ch. 257, where Karasuma revealed that he knew about Tenma's feelings and felt the same way - and yet at the same time knew that his condition made it impossible. Not only that, but it implies that Karasuma put off having surgery for a whole year, just for Tenma's sake. Unlike Harima, Karasuma's feelings are not possessive in the slightest - his desire to stay is driven by his need to see Tenma a little longer, and his wish to tell her that she's beautiful. It's up to Tenma to decide what to do with those feelings - Karasuma says that he will be fine with whatever she chooses.
The sidetext observed that Karasuma's buried feelings are coming to the surface, too; Harima isn't the only one who wants to get rid of pent up frustration. Also, I don't think you are being fair to Harima here. Lashing out physically is his only way of expressing his feelings, and Harima is feeling hurt and betrayed by Karasuma. Harima has a great deal of respect for Karasuma. After stepping aside and letting Tenma go to Karasuma, Harima's trust was seemingly betrayed by Karasuma. Harima is driven by his (implicit, not explicit) rejection by Tenma, but those feelings are focused into rage at Karasuma, who threw away Tenma's feelings and hurt her.From the very start, Harima's fundamental mistake is that he views Tenma as a possession that he can somehow obtain. He feels that anything concerning automatically concerns him, which is why he feels justified in confronting Karasuma despite the fact that Tenma's relationship is her own business. But as is pointed out in Ch. 254, Harima really just doesn't understand Tenma. Tenma is free spirited - she does whatever she wants, and loves whomever she wants, and nothing that Harima says or does can forcibly change that. Chances are, if Tenma finds out that Harima is assaulting the one she loves, she probably won't be thanking him for it.
The biggest problem is that Harima somehow feels justified for starting the fight. But even as he plays the role of the hero ("Love her enough to die!"), his actions are overflowing from his own feelings of frustration at being rejected by Tenma. As grand as Harima's gestures are, he is ultimately only trying to make himself feel good by lashing out at Karasuma. Karasuma, too, understands this, which is why he agrees to fight Harima if it will help Harima get rid of his pent up frustration.
Looking at it from a literary perspective, confronting Karasuma is the most heroic action for Harima. His alternatives are to do nothing (which is right out) and to try to comfort Tenma himself. However, comforting Tenma is not something that Harima is in a place to do; it would also be dangerously close to moving in on her in her moment of weakness. Confronting Karasuma and punishing him for hurting Tenma or even trying to force him to apologize or accept her confession is something much more within the scope of Harima's abilities.