AHC: Implausible Classical Age of Exploration Idea

This is one of those ideas I come up with while lying in bed. It is a twist on the 'what if the Romans colonized the New World' scenario that never really stands up to much scrutiny. Your challenge is to take the premise of my idea and make it more plausible.

It starts with Alexander's Empire, insofar as it controls the Eastern Mediterranean, never collapsing as badly as it did historically, so that their borders resemble the Byzantine or Ottoman Empires. In other words, Greece, Anatolia, the Levant, and Egypt, at minimum. This gives them control over trade with the East. Meanwhile, Rome and Carthage never come to the point of wars of annihilation, and, while they may be generally hostile toward each other, they co-exist in some fashion. Carthage conquers Hispania, Rome conquers Gaul (meaning both have unrestricted access to the Atlantic). With three major powers in the Mediterranean, naval technology gets some more focus. At some point, sailors (likely Carthaginian) discover the New World, while trying to circumnavigate either Africa or the Earth.

Carthage and Rome now can compete for colonies in the New World. Bonus points if a Celtic Britain gets in on the game, to compete with the Latin and Punic colonies.

Alright, with that set up, how can we make this believable? The smaller the changes from the basic premise, the better.
The biggest problem for me is the lack of incentive. The classical world doesn't need to go west to trade with china like the medieval Europeans felt the need to. The new world certainly has its unique resources and abundance of traditional ones they aren't going to be known immediately. Potosi isn't exactly on an island off the coast of Brazil
Carthage discovers the Azores soon after their conquest of Hispania and then for whatever reason explore farther and farther west until the reach Bermuda. After a few years of settlement on both, their sailors continue west until they reach the Bahamas putting them in range of the Carribbean. You'll need one hell of an incentive for such journeys though. Religious revelation perhaps?

Celtic Britain can reach America via the Norse route of Iceland>Greenland, both are unpopulated at this point OTL, so a simple population surplus can be used as incentive for westward exploration. That said neither have great arable land so they'll need to reach America rather soon after each discovery before they're wrote off as unsuitable.

Romans can sail direct to Newfoundland from the tip of Brittany, Newfoundland is unpopulated at this point OTL afair, so they'll have further incentive once they reach the Grand Banks. The initial incentive, again, is the problem for such a journey.
I guess IF the Alexandrine state acts like the OTL Sultanate of Egypt and blocks anybody else from going East via Egypt to India and China, then the others have a purpose in going West, and once they discover somewhere that is NOT India or China but something entirely new, then Alexander's state would see a reason to do so too