This interests me. I'm going to provide my full thoughts on the matter.

I don't think Rinne makes an intentional effort to rage bait Keito most of the time, if that makes sense. Like obviously he mocked AKATSUKI in Main Story 1, but in the day-to-day, I think it's more like: Rinne does something that violates the rules of Seisoukan or is otherwise disruptive, Keito reacts negatively, Rinne responds mostly neutrally and maturely to being corrected, as well. At least, that's how I see their relationship working. Comparatively, Ibuki does go about teasing Keito on purpose. Though, it's worth noting most of his "troubling behavior" is not an effort to get Keito's attention on its own. The teasing usually is a reaction to being scolded. It's his way of taking control over the situation again. But, of course, it'd be disingenuous to discount Ibuki's enjoyment of poking at others lol. So, that's basically how their rhythm works, and they grow used to it. Ibuki is, therefore, more experienced at rage baiting Keito. Their dynamic also means Keito is not hostile when Ibuki tries to push his buttons. This means that, while Ibuki knows how to get on Keito's nerves better, the reality is that Rinne is more likely to actually make him angry if it became clear he was intentionally trying to provoke Keito.

But the real question is, would Rinne and Ibuki partake in such a competition? And to that I have to say, perhaps surprisingly, no. To me, it seems more like something Rinne would suggest after observing Ibuki and Keito's interactions. Maybe he also heard from Hiiro about what happened at the Lecture Circle lol. Not that Hiiro's a gossip, but he's definitely a concerned friend regarding Ibuki who might want Nii-san's advice after a pointed complaint from Aira comparing Ibuki to Rinne. At any rate, I think Rinne would find the way Ibuki teases Keito amusing. But tbh, I don't think Ibuki would accept such a proposal as a contest to see who can annoy Keito more. I think he'd get serious all of sudden and say "Don't cause problems for Hasumi-san." Rinne would be shocked. He'd go whine to Rei about it later when they're out drinking:
"Can you believe Bukki scolded me like that, Rei-chan? About causing problems for Keito-chan, no less..."
"Hm... Those two seem to have a special relationship. I think it's best to not interfere."