Military orders (like Knights Templar) were only established as a result of the First Crusade to secure pilgrims to the Crusader kingdoms and wage holy wars, which hadn't really been a thing before the First Crusade. This means that the practicing priests among the crusaders were not part of the military, they were there doing priest stuff, not swinging swords. So my headcanon is that Nicólo is originally from a landlord family and was trained as a knight, but he didn't want the heterosexual feudal lifestyle so he joined the church. And when the First Crusade was called, he left the church and joined the crusade as a knight.

Genoa did participate in the Crusade but with a navy, not with an army, so I wouldn't make sense that Nicólo would have fought under Genoan banner, especially if he died in the Siege of Jerusalem as it's implied. So my headcanon is that he joined the army of Tancred, one of the two Norman led Southern Italian armies which were part of the Crusade. Maybe he was in Rome as a priest to study theology or something when the crusade was called.