I doubt that Titus was against the act itself.
http://www.eyewitnesstohistory.com/jewishtemple.htm
Colosseum ‘built with loot from sack of Jerusalem temple’
http://www.telegraph.co.uk/news/worldnews/1311985/Colosseum-built-with-loot-from-sack-of-Jerusalem-temple.html
Israeli Rabbis Hope to Search Vatican
http://www.freerepublic.com/focus/news/1058869/posts
Actually Josephus is where I got the idea that Titus didn’t want the Temple destroyed. This is from the Josephus that you linked to:
“As they drew closer to the Temple, they pretended not even to hear Caesar’s orders, but urged the men in front to throw in more firebrands.”
“When Caesar failed to restrain the fury of his frenzied soldiers, and the fire could not be checked, he entered the building with his generals and looked at the holy place of the sanctuary”
“As the flames had not yet penetrated to the inner sanctum, but were consuming the chambers that surrounded the sanctuary, Titus assumed correctly that there was still time to save the structure; he ran out and by personal appeals he endeavored to persuade his men to put out the fire, instructing Liberalius, a centurion of his bodyguard of lancers, to club any of the men who disobeyed his orders. But their respect for Caesar and their fear of the centurion’s staff who was trying to check them were overpowered by their rage, their detestation of the Jews, and an utterly uncontrolled lust for battle. “