Roman Republic isn't in the game… Augustus represents the Empire.
As far as UA, Rome's "strength" (I hate their UA) is infrastructure, so Order is best suited. Of course, Order enables your cities to build everything easily no matter who you are, so it's not really in need of any UA synergy. And Autocracy likes to buy everything. Freedom, which lacks as much raw production or gold, but encourages tall, capital-first play, actually probably synergizes best with Rome's UA. Even if you go wide, as Freedom can work for wide if you staff 5 specialists in every city.
Historically/politically, Imperial Rome has resemblances to all three. Omnipotent leader, rampant, expensive elections, slavery, grain dole, and world domination - what ideology don't they match?