Arthur was still underage when Richard died, killed while trying to put down one of the revolts among the continental barons endemic to the Angevin Empire. With Richard immediately returning in triumph (and having the added prestige of helping to actually take Jerusalem to boot) rather than having to be ransomed back from an archduke he insulted after spending two years in captivity, I think there's a good chance that
that particular rebellious lord might choose some other time to rebel instead, and buy Arthur a few more years to mature.
Yes, it'd be a hell of a challenge, but then so was creating the Latin Empire and associated crusader principalities atop the temporarily-mostly-dead carcass of Byzantium. The crusaders will be further aided by the Ayyubids having just come out of a civil war between Saladin's sons and brother, not
that dissimilar to the turbulence of the Angeloi period in Byzantine lands. I also don't think it's a given that events in the ERE will play out exactly like they did historically with 5-10 years' worth of butterflies, it took a very specific chain of disastrous events for that to happen - just not having Alexios III succeed in deposing his brother Isaac II while the latter was on a hunting trip in 1195 will derail this course, and put off some sort of fatal Byzantine-Latin confrontation (which admittedly was very likely to happen at some point after the
1182 Massacre of the Latins) for a while longer.