The reasons were already mentioned and gone over (North African Christianity being a mess and poorly established where it would need to be--Mauretania, the late introduction of the camel and the fact trans-Sahara trade was in its infancy, etc.), but I think the best is definitely getting camels introduced earlier and trans-Saharan routes more established. That, and have the Romans conquer deeper into Mauretania. While it isn't exactly the best of land, it also isn't terrible land, either, and I think its doable for the Romans to at least nominally subjugate most of modern-day Morocco, and establish firmer control over far more of the region than they did OTL. More military colonies could be established too in Mauretania, especially Mauretania Tingitana. If Mauretania is an integral part of the Empire, then its far more likely Christianity will spread on the trans-Sahara routes and convert Ghana.
Since this would be happening early on (the late empire won't do it, they'll only reconquer it if its lost and even then I wouldn't be sure), you'd probably butterfly Islam, leaving Christianity in control of the Mediterranean. I think the common way of converting a kingdom--the whole "saint(s) go in and convert the king" stories--is unlikely to occur in West Africa based on West African traditional religion. There's no way they'd be submitting to any spiritual authority since the king himself was basically a god (incidentally, that is a reason why the kings took a long time to convert to Islam). Caesaropapism is always a way around that, but also intriguing is maybe get the king to simultaneously be the head of the church in his region--"king-archbishop", "king-patriarch", or something, and obviously with no concept of clerical celibacy. It goes against traditional Christianity, but African Christianity is hardly likely to be traditional.
One idea I was toying with if you get Donatist or other heretics crossing the Sahara is the proliferation of the Tifinagh alphabet. It's a long shot, granted, but since the alphabet was in use until the 3rd century AD, and mainly amongst Berbers and Punic speakers (who seem to have been disproportionately represented in heretical sects), they could cross the Sahara and note the Tuareg use of the script (and hopefully convert a tribe or two of Tuareg. From there, they use it to write down the languages of the sub-Saharan African peoples. Why Tifinagh and not Latin? I'm not sure--but since Berber and Punic are both Afroasiatic languages, wouldn't Tifinagh be a better choice for writing a language like Hausa or another Chadic language? Granted, that would only be for the Hausa states and no one else, but if the Hausa become the first literate sub-Saharan African society, then that might make help spread the language. I know realistically Latin is more likely to be used if any sub-Saharan African languages are to be written down, but a potential Tifinagh connection is too good to pass up. A Bible translation made by these heretics into Punic (somewhat surprising one never existed OTL) would be interesting too, in getting Punic used as a language of administration in West Africa (though Punic had been written in the Latin alphabet since not long after the Roman conquest).
Maybe if some of the Donatists ( or possibly other heresies ) escape southwards to escape persecution ? Trade routes between West and North Africa existed so this might happen, and they would have a few centuries to solidify like in Ethiopia, though the butterfly effect might mean no Islam thus no huge challenge to Christianity in the region.
It wouldn't be in Ghana, though, since Ghana was linked to Mauretania/Morocco by the Trans-Saharan routes. The most immediate from Donatist strongholds in North Africa would be near where Kanem-Bornu and the Hausa states would emerge. But it isn't totally out of the question that North African Christians would flee--and we know that indigenous Christianity in North Africa lasted until the 13th century and possibly even later.
I don't think Christian refugees would really work, at least not in converting the ruling classes. They have no prestige attached to their religion and they have no connections with the outside world (a Christian North Africa is absolutely necessary for Ghana or other sub-Saharan Africans to convert). They'd have extreme difficulty convincing the ruling class to convert, when that was driven by pragmatic reasons. The countryside would also likely not convert in any significant numbers.