Very, very high chances of surviving. It even could've survived under Islam if the Donatist schism had produced a church as organised akin to the Copts or Syriac churches were. If that had happened, Punic would've survived at least into Early Modern times (like Coptic), if not to this day (as many Aramaic languages have). The biggest disrupting influence seems to have been Vandal rule which broke the continuity in Roman rule since the fall of Carthage. There's also Arabic sources that suggest it was spoken into the 10th/11th century, which wouldn't surprise me, since in Augustine's time it was the majority language in North Africa even if basically everyone also spoke Latin.
And I think without Islam, it almost certainly will survive as a minority language, at the very least, for another few centuries. It seems like that by Late Antiquity it had evolved a certain association with schismatic branches of Christianity (Donatists) but also Manichaeans (St. Augustine was a fluent Punic speaker and ex-Manichaean), though evidently not the Arianism promoted by the Vandals. For maximum success, get the language associated with a proto-nationalism (associated with either Donatists or another church schism) as soon as possible, maybe by having the Exarchate massively implode. Because the language does seem to be declining post-Vandal from what I can tell (especially in coastal areas, where it seems to have been slowly declining by the late 4th century based on epigraphical evidence), and religious orthodoxy (North Africa will look to Rome, of course) will lead to continued linguistic Romanisation and expansion of African Romance.
Are there any resources online that tell of the language's changes under imperial rule or of the typical names used by Punic-speakers?
Late Punic Epigraphy: An Introduction to the Study of Neo-Punic and Latino-Punic Inscriptions has a sizable free preview online, but I'm looking for that as much as you are. The language itself has very significant changes from even early Imperial times by Late Antiquity. Many, many Latin influences. Punic speakers tended to use Latin names for the most parts, rendered in Punic--something like "Severus" would be "Sewere" if I recall. This was especially common by Late Antiquity. There was also huge Berber influences that were gained throughout Roman times when Punic displaced Berber languages/Numidian from a lot of Libya, Tunisia, and parts of Algeria. I find it amazing how the Punic language is proven to have
expanded under Roman rule. This is why a lot of them use Libyan/Berber names too. It seems like Semitic names were relatively uncommon, though.
Oh, it also seems like the /p/ in Late Antiquity Punic had been lost, possibly by prolonged contact with Berber languages or natural evolution.
