My initial impression of Carthage is that it's very weak even with Augustus. The unique buildings are really bad investments in towns unless you can get a lot of things going to bring the value of those buildings while reducing their costs. A 6-production Cothon is a good investment in the capital as it costs 120 production. If you ignored its maintenance, you'd break even in 20 turns, which is excellent. In a town, the 6 production becomes 6 gold. When you take away 2 gold from maintenance, you net 4 gold per turn, and it costs 480 gold base to purchase. With Augustus' bonus, the cost comes down to 360 gold with a breakeven period of 90 turns.
Here's my hypothesis on how to get the most out of Carthage. I haven't actually tried out all of the things I outline below.
To make Carthage work, you have to prioritize two things. First, you need to get as many copies of Gold as you can for the +20% purchasing power per copy. Second, you need to befriend some city states. In particular, you want 1 science city state and 2 gold city states. For city state bonuses, you want to select free tech for each suzerainty, +5 trade route length per city state, and +1 to gold buildings per city state.
The science bonus is needed because science per turn is hard to get with Carthage. You should take a free tech whenever you can. Keep in mind the free techs you get this way are often masteries, so you want to leave things like Masonry II and Currency II un-researched in anticipation of future suzerainty even though these directly improve the value of the unique buildings.
Beelining the Carthaginian civics is a bad idea because the unique buildings have terrible value at the start of the game. You want to beeline Code of Laws instead and get three merchants out (get one by unlocking Code of Laws and build one more and get one free). I think you get an economic attribute point for establishing two or three trade routes? There's an early narrative event for Carthage that lets you choose between economic and military attribute points, and so if you start the game with the memento that gives you an economic attribute point, you can unlock two key attributes early on: +2 gold per trade route, +15% purchasing power.
For the other memento, most people's pick seems to be Corona Civica for +1 settlement limit, but because of Carthage's dependence on city states, I think going for a free diplomatic attribute point is perfectly viable as well. I don't know if Carthage gets a diplomatic attribute point for befriending 1 city state, but if they do, you can use that point for +50% toward befriending city states. Otherwise, building Emile Bell is an option, although I think that's a little expensive since it requires Code of Laws II and further delays working on the Carthaginian civics.
Another thing that's key to getting the most out of Carthage is to get the narrative event that gives +1 influence to each settlement. I'm not really sure how this is triggered, but I have a suspicion this has to do with the order in which you unlock the Carthaginian civics. I was testing out beelining Cothon in several games yesterday and I believe I never once got that event.
As Carthage needs to lean heavily into influence generation, I think Machiavelli might actually be a decent pick. Machiavelli's both diplomatic and economic, so you're getting all the attribute points you need with him. +3 additional influence per turn is very useful, and Augustus' bonus of +50% toward town buildings is somewhat offset by getting gold whenever you offer a diplomatic action. But then, you're also missing out the ability buy monuments in towns as Augustus, and before you can (or should) buy the unique buildings in towns, monuments are probably the best investments with the gold you get.
