So I get that Obito was disillusioned with Kakashi because he didn't keep the promise of protecting Rin. It was Obito's greatest desire to see the girl that he loved, for which he risked his life to save, to keep living on, and he entrusted Kakashi to make sure it happened but Kakashi wasn't able to.
I'm starting to think that Obito also became disillusioned with Rin as well. He might feel betrayed by her because despite knowing how much Obito wanted her to live on she instead chose to die, therefore denying him his desire as well.
So I think Obito might somehow feel betrayed by both Kakashi and Rin, so he claims them as fakes since in his mind the real Kakashi and Rin would've honored Obito's last wish.
Maybe Obito felt that Rin valued her love for Kakashi more than his love for her, therefore choosing to die at the hands of who she loves instead of trying to find a way to live for the sake of who loved her. In a way making it seem that Kakashi was more important to her than Obito ever was.
Maybe a way to redeem Obito would be to reveal that Rin was taking Obito's feelings more into consideration than he initially theorized. Maybe her resolution to die was actually inspired by Obito, or something like that. It would be an interesting development. And that way it could make Obito not consider the Rin that died to be a fake anymore, and thus decide to honor her memory somehow.
I got an idea. I think the best way to convinced Obito is to let Naruto do the talking. Why? Think about it. Kakashi already established that they are similar in many ways. Kakashi thought to himself it's never too late because he can still change, but Obito is on the verge that everything he sees are fakes. So fake, that his heart is no longer there. Genjutsu or not, the message is that he has no heart to give in hope for. Kakashi can't convince him or maybe he can, but let me get to my idea. The thing is that Obito can change if only he got the same time span to talk to Naruto, one-on-one, like Nagato. Naruto represents the future, so perhaps Kakashi can prove Obito that things will change and it's already progressing.
Naruto is known to relate his enemy and change his enemy to die as a human, not a monster. Obito is about the past ninja lifestyle while Naruto is about the future repenting the mistake of the past. Remember, Naruto hates the law of ninja with people die at all cost rather than protect them and see another day with your friends, which Obito once believed. I think his breakthrough should be about understanding that life has changed and Naruto is the one. He denies it, but perhaps he wants to see it for himself one last time. Maybe Kakashi could convince him that Naruto is the one to fix all and Obito will play one last game by either he wins by killing him or loses by Naruto. I should not reveal too much, because I feel like Kishi reads this and swerve it to make it unique, which may result in "Huh?" but that's just me.

Plus, I do want Naruto to conclude with Obito because they really had a feud going and if Obito just come back good, he'll be like "What the hell happened?" and it will be less satisfying. Not to mention, he does have to do for people of present and his parents. It seems given that Naruto will confront and end Obito, but who knows. I think it's best to have them settle the score. Right now, it's Kakashi, but I don't think it will end with him. Kakashi can bring up Rin's truth and such, but I do think it should happen when Obito is like 85% done of converting by TnJ.