Like in the classic 'Rome sweet Rome' scenario, a MEU is magically transported to Rome, with the following differences: They arrive at the time of the insane emperor Caligula, they are from the first second aware of the situation, and they know that they are in a parallel universe, so they don't have to worry about causality loops, (Whatever they do, they can not erase themselves by the removed consequences changing the modern world.) and they happen to have a man who speaks Latin properly and is highly educated in ancient history.
Caligula is not too clever, but is violent and holds himself a god, and he attacks the Marines with all he has. (Praetorian Guard and Urban cohorts) In the resulting battle the Roman forces are destroyed, the Emperor, many of his retinue and the Praefectus Praetorio is taken captive. The next day the Marines attack Rome, meet no resistance, and seize the Palatine Hill and the Capitol, and capture many senators and the Praefectus Urbi, and the members of the Julius-Claudius dynasty, including Claudius himself.
However, the the CO of the Marines knows that they can not control the whole Empire, nor stand against it indefinitely due to the lack of supplies and manpower. Becoming a bit dreamy from his power (and knowing it is parallel universe) he decides to transform the Empire as much a possible by his resources, according to his ideals. The Marines want to: (not in this importance or time ordering)
1) Promote human rights, including the abolition of slavery
2) Institute some kind of democratic government, building on roman republican tradition, but making it better than the original republic that had collapsed just some decades ago.
3) Give the provinces equal rights with Italy
4) Minimize own and roman causalities
5) Prevent the empire from falling apart, collapsing or sinking into civil war.
6) Prevent persecution of Christians
7) Provide some form of retirement for the Marines after years long labor in the Transformation.
How is this to be achieved?