You should not weigh too much on their religion's effect on political relations. Of the "Catholic" countries of Europe, France and Italy were overwhelmingly transformed by the 18th-19th century revolutions and are aligned with French revolutionary ideals which are strongly anti-religious (laicite). Among the others, religion is only important in Spain and Poland, where Catholicism continues to be a nationalist rallying point (for Spanish, their hatred of the French and separatists, and for Polish, their hatred of Russia). All the rest, where the religion no longer has a welfare or nationalist function, are completely indifferent to religion. The situation is not at all like US-style Evangelicalism who are an entirely new politically-minded religious revival that appeared in the 19th century and never took hold in Europe.I highly doubt the Protestant and (especially the) Catholic Europeans will ever develop any kind of cordiality with the Orthodox Russians. This hostility is as old as the Great Schism and Division of the Roman Empire, with Russia essentially being the legacy of the Orthodox Byzantines. For example, it was only a few years ago that the Catholic Pope officially apologized for the Catholic Crusade in the middle ages that backstabbed and irreversibly devastated the Orthodox Byzantines.
You brought up the Sack of Constantinople because the Orthodox bring it up very often. But that's because the Orthodox countries are much more religious than any western Christian state. Virtually nobody in the West still brings up the fact that the Sack of Constantinople was revenge for the much more brutal Massacre of the Latins by the Greeks a few years beforehand on top of the Greeks refusing to pay their debts (scroll down that Wiki article if you dont know about it), because the modern "Catholics" just don't care enough to even bear a grudge (or the French, Italians, and those aligned with their views, who would blame religion in general)
And even the Orthodox don't have a common approach to the "Catholic" countries. The non-Russian part of Orthodox Ukraine are insane Polish worshippers. Orthodox Bulgaria is the most pro-EU country in the Union. Georgia, North Macedonia and Montenegro love getting help from the West to cause trouble with their fellow Orthodox neighbours. Only Serbia, Russia can be firmly counted as anti-Western because national identity is much more influential in political decisions than purely religion.