I'd concentrate on the naval theatre then. In those situations I just blockade and cut off their income as much as possible. No gold, no ability to wage war.

You could get a submod to reduce the AI spawning too many armies if you feel it is too unrealistic.

Sounds like a fun campaign : ) (Though a bit typical : P )

You first need control of the seas. You can't beat carthage otherwise. Make sure they have no fleets going around. Blockade their capital, kill fleets while they are recruited.

Just keep defending until you build your economy up.

When you feel confident and can afford an extra army, get a foothold in Africa. Not close to their capital, or you will get swarmed by all their armies.

My personal fun times in Sicily is farming the armies Carthage throws there. If you let one settlement stay undefended you can retake it (and loot it) every time Carthage has taken it, rinse and repeat. Different ways to do it of course, but after Sicily fun I honestly like to take Carthage capital first, due to how you need multiple armies and navies to defend Sicily anyway and that is just a very short cross over the over sea. I actually think you can use Akragas port to move the armies over water in one go to attack Carthages port due to how short the sea distance is there? Lacking their capital city should cut down a bit their ability to create navies and armies. Holding that wont be hard and once you ready have easy access to also destroy their client states nearby.

And a lil bonus: Now I dont know if this is actually an exploit, but if you attack Carthage city from the port then there is a hole in the wall on the backside (only accessible when attacking from the port). It has no gate and no overarching wall (so no boiling oil to throw down) which makes it quite a lot easier as you can pelt their units that fill the gap with missiles.