The Roman interpretation of the Carthaginian pantheon is a fascinating topic, because was affected by a number of factors including the unique Roman attitude towards religion, Rome’s interaction with their empire and former conquered states, and finally Roman trauma towards Carthage and the Punic wars.

First, I’ll briefly touch on Rome’s attitude towards gods and particularly foreign gods. Rome’s general attitude was to avoid risking the wrath of the gods and therefore treat them and their local religious practices with what they believed to be the respect they desired. This meant obeying local customs under the theory that gods may have particular ways they wanted to be worshipped in particular locations. While I wouldn’t call it making “contracts” with the gods, the Romans were careful to put forward a ceremonial message that they were trying to satisfy the local gods with their actions. At all times, the Romans wanted the gods on their side, and not their enemies.

As an example, as the final part of their war against Veii, the Romans called upon the goddess and made the following statement: “Juno Regina, you who now inhabits Veii, I pray that you will follow us as victors to our city, soon to be your city, where a shrine worthy of your majesty will receive you.” (Livy 5.21)

Basically, the Romans are bribing the goddess to switch sides.

In the case of Carthage, this means that the Romans are going to mimic the practices of the Carthaginian gods when in Carthage, in the way that they think is correct.

This can be called Romanization. It also involves the conversion of certain names to ones the Romans understand and place them within their own pantheon. For example, the most important female goddess of a region may get called Juno (this is why Tanit gets called Juno Caelestis). The Ugaritic god El is often called Kronos. In the case of Baal Hammon, the alleged child sacrifice as part of the tophet is linked to the myth of Saturn eating his children, and therefore Baal Hammon gets called Saturn.

Marcel LeGlay pioneered the use of the term “interpetatio romano” to describe the transition from Baal Hammon to Saturn. He suggests that Saturn Africain continued to be a more “orientale” [sic] god than Hellenistic, and that the main change was just in the name of the god. He states that Saturn Africain had a Roman identity, but the personality of an African god, and was “absolutely not Roman”. While “interpretatio romano” is not the ideal term, it does a good job of describing how the Romans kept the essence of the worship while practicing in a way more amenable to their sensibilities.

As Saturn in the Roman period, many elements of worship remained the same, such as the style of vows on stelae and the existence of particular sites of worship. However, there were shifts in certain trends, such as the move towards rural, open air sanctuaries and the use of “substitution sacrifice”. In Punic times, temples to Baal Hammon were sometimes inside cities and sometimes on the outskirts. After the Roman rebuilding, this trend accelerated and nearly all of the major sanctuaries are located outside cities or far from cities in the mountains. Some of these sites include Thugga, Timgad, Uchi Maius, and Thinnissut. The biggest site was mount Balcaranensis, which was so important that one of the most common names for the new Saturn was “Saturn Balcaranensis”.

The other major trend is “substitution sacrifice”, and I think this is the most interesting. Whether the tophet actually involved the murder and sacrifice of live babies and toddlers or if it was miscarriages and/or dead babies is still hotly debated. The Romans appeared to believe the former. Considering their hatred of Carthage and the role of the tophet in anti-Carthaginian propaganda, it is easy to understand why the practice of infant sacrifice did not appear to exist in Roman times. Instead, votive stelae from this time period show images of animals (such as bulls), indicating the sacrifice was changed to animals. However, other aspects of the practice continued. Part of the tophet was an apparent contract with the god, and phrases such as “Caius Iulius has executed” or “Lucius Terentius has accomplished his vow” appear on stelae from this time period. Note the names are very Roman- these are Roman people conducting a Carthaginian religious practice!

The Romans did not discourage the worship of Saturn in this time period, as theoretically “Saturn” is a Roman god. Instead of being called Baal Hammon, he became “Saturnus Augustus” by the 3rd century CE. Saturn had the most temples dedicated to him by private citizens of any god in Africa (Rives, Religion and authority). There are records of the Roman government giving money to restore temples believed to be dedicated to Saturn, which is particularly noteworthy because usually they only did so for the Capitoline triad.

To summarize your question: Did Punic paganism survive into Roman times? Depend on how you define it. The names of Baal Hammon and Tanit did not- they were rebranded and reformed into Saturnus Augustus and Juno Caelestis. Did their practices survive? Yes, elements of the tophet continued well into the Roman Empire in the 3rd century CE and later. However, this was not a case of Punic descendants quietly worshipping their own, separate pantheon in secret. The worship of Saturn and Juno was conducted in the open, by Roman people, alongside the Capitoline triad and other more classically Roman gods.