I don't think you can avbouid the schism after 1000, when it is pretty much reality and the papacy begins to assert its power in all directions, soon to begin making an issue of the differences. It might have been bridged in the 14th or early 15th century, but only formally. I doubt after 1204 many Byzantines had much love for the Latins. After 1453, the Ottomans may have been too wary of a large group of their subjects suddenly swearing allegiance to the Pope...
But a much nicer way would be for the Western church to stop acting like babies insisting on 'running their own show' (I mean, where would *Rome* get authority over *Jerusalem* and *Antioch*?). If we could get Rome to accept the Eastern version of Christology at least as equally valid, or essentially the same thing (not that hard, really. Few people understood or understand what it is actually about, and barring some fool telling them their eternal salvation hangs on it, I doubt they care. Most doctrinal disputes are solved without street warfare.) that'd be a promising beginning. The next step would be to stop mandatory celibacy (voluntary celibacy, as existed up until the 12th century, is fine. It's the mandatory bit that is a problem, and that was instituted largely fpr monetary concerns). Then, the 'Pilatians' should lose out in the missionary dispute in the East, allowing liturgical languages other than Hebrew, Greek and Latin (there's nothing in Catholic dogma that actually forbids it, but it was a hugely divisive issue along the religious frontiers because it is so obvious). That might have the added positive effect of allowing the survival of vernacular literatures in Old Prussian, Finnish, and Wendish. That would take care of the issues that require large adjustments to iron out - married priests, masses in Slavonic - and the rest is esoteric doctrine.
Remember, the Catholic Church may have a very different look and feel to the Orthodox, but that is mostly cultural baggage. On Sardinia, Catholic clergy practice the Catholic Greek Rite, which is almost indistinguishable from Greek Orthodox. They don't marry, and of course they are lucky enough to have Greek as their liturgical language (it wouldn't work with Russian or Serb Orthodox communities), but all other issues have been settled without any trouble.