I think you've overestimated the power the Pope had to command Catholic nations, and underestimated how close the two churches actually got during Byzantium's final century. The emperors of Byzantium were literally begging Catholic nations to help stem the tide of the Turks (John V Palaiologos even went so far as to offer submission to the Catholic church in the 14th century). But, who could come fight, even if the Pope commanded then to? England and France were still on the road to recovery from the 100 Years War. The Holy Roman Empire was dealing with internal upheavals. The Battle of Varna in 1444 utterly destroyed Hungary's ability to wage war, and killed off any Crusading desires in the west.

However, Byzantium's calls did not go completely unanswered. Genoa and Venice, both, contributed to Constantinople's defense, and several thousand foreign mercenaries from across Europe volunteered to fight as the bulk of the city's defense. The sad truth, though, was that most of Western Europe simply did not care no matter how much the pope demanded, and Byzantium was already in a state of downward spiral since the Fourth Crusade, which it could never recover from. It's defeat was genuinely inevitable, and the fact that it lived as long as it did and defended Constantinople against the final assault in 1453 as well as it did was nothing short of miraculous.

Source: J. J. Norwich, "History of Byzantium."

"Fully defending" Constantinople required troops, ships and money that the Pope didn't have and could not command, leaving aside that the Byzantine 'empire' itself had effectively shrunk to the city and the Morea and thus the Ottomans could simply have returned at any time.

Furthermore, by that point Constantinople was of little strategic significance - the Ottomans could move troops from one side of their empire to the other at will, as the events of Varna itself showed.

That the pope was well aware that commitment to the Union of Florence was not that deep was incidental.

But they did!

The Pope pleaded to the Catholic nations of Europe to go and help the Byzantines.

The problem was the schism and the anger that had developed between the Byzantines and the Latins, between the Orthodox and Catholic, had gotten even worse by the time.

It had gotten so bad that some Orthodox openly said that they would rather submit to an Islamic Sultan than they would to the Pope.

Funnily enough, the fall of Constantinople is what finally fired up the Renaissance as the fleeing Greeks came and settled in the Italian city-states.

Also, historians mark the fall as the beginning of the "Modern Era" - essentially, the Middle Ages finish in 1453!