Okay, so the first thing to say here is that this is a period of church history that is not particularly well-documented. James Fraser addresses the issue of southwestern Scotland's patchy representation in the sources in his article "Strangers on the Clyde: Cenél Comgaill, Clyde Rock, and the bishops of Kingarth" (Innes Review vol. 56 no. 2, Autumn 2005, pp. 102-120). (Clyde Rock is an alternate name for Alt Clut, also known as the Kingdom of Strathclyde; Alt Clut is also a name for Dumbarton Castle.) The problem is the overwhelming dominance of Iona in our sources. Fraser's article explores how the toponymic, archaeological, and fragmentary historical evidence suggests that there were significant ecclesiastical powers in the southwest which barely merit a mention from Iona's chroniclers. Part of this, Fraser suspects, is that Iona remained conspicuously silent on some of its greatest rivals. That leaves us with the difficult reality that some of the most important ecclesiastical players were being purposely downplayed in our sources. Although his article primarily concerns the 7th century, I think you'd find it very useful for coming to terms with what we know and, more importantly, the limits of what we can know about the religious landscape in this area.

Next I'd like to address your question about the "schools of Christianity". There were no major doctrinal issues dividing the churches of Ireland and Scotland during this period. The conflicts over the calculation of Easter and the tonsure had been settled in Scotland the 8th century at the latest. Even then, those were not different "schools" of Christianity per se, just a conflict of institutional traditions which had to be reconciled through royal decree throughout Britain and Ireland in the 7th and 8th centuries. The main diversity at play would have been the proliferation of different monastic rules. However, while the surviving monastic rules from roughly 8th-11th century Ireland often show local inconsistencies, it's not clear that any of the monks in these places would have considered others to be following a different "school" of Christianity than they were. Given the diversity in the sources, it may well have been expected that different communities would follow slightly different rules, since 870 predates the widespread adoption of the Benedictine Rule in either Ireland or Scotland which standardized monasticism to a degree previously unknown in the West. If you'd like to read some of these rules, check out the book The Celtic Monk: Rules & Writings of Early Irish Monks translated by Uinseann O' Maidin (Kalamazoo, 1996).

The main exception to this was the rise of the Céli Dé (sometimes Anglicised as Culdees). The Céli Dé ("servants of God") were a particularly ascetic group of Irish Christians who spread to Scotland. They were on the rise in Ireland throughout the 9th century, and Thomas Clancy has suggested they may have been in Scotland in the 9th century too (see "Iona, Scotland, and the Céli Dé" in Barbara E. Crawford (ed.), Scotland in Dark Age Britain (St Andrews, 1996)). The most famous Scottish institution of Céli Dé was in St Andrews, where Clancy argues in the above article that the Céli Dé had established a church by the mid-9th century. The Céli Dé could be described as a different school of Christianity from the mainstream, but their differences were of practice rather than doctrine. There are no known heresies originating in either Scotland or Ireland in this period, so while there would have been plenty of strong opinions about which monastic house had the best practice, these can't be characterized as serious doctrinal conflicts. One of the main characteristics which set Irish and Scottish Christianity apart from forms on the Continent at this time, however, was the practice of having a soul friend, or anam cara. If you read The Celtic Monk, recommended above, you can see what the responsibilities of a soul friend entailed. It was something like a confessor, and the Irish penitential sources show the development of confession as a lifelong, habitual practice, rather than the one-off event it had been in the late antique church.

By 870, the religious landscape of northern Ireland and southwestern Scotland had been seriously shaken up by Viking raids. The traditional dominant powers of Armagh and Iona had both been raided. Iona in particular was in serious decline by 870. The relics of Columba were removed from Iona and dispersed to different monasteries such as Dunkeld, which is further east in Scotland. This was also a time of major political shift in Scotland. Dàl Riata and been conquered in the 8th century and, as I'm sure you're aware since you're researching the Viking siege of Dumbarton, Alt Clut was on its last legs by then as well. After the battle, Govan seems to have filled the vaccuum and become a major church centre, but it would still have been a relatively minor monastery in the time leading up to 870. The Hebrides were starting to undergo serious Norse influence at this time so the Church in these areas would have been in serious decline.

Above all, though, the late 9th century saw a shift eastwards as the Pictish and Dalriatan kingdoms unified into the kingdom of Alba. Eastern church sites such as St Andrews grew in importance during this period; our first version of the St Andrews foundation account dates to this time (for which see Simon Taylor and Gilbert Márkus's Place-names of Fife, volume 3: St Andrews and the East Neuk). So the period around 870 would have been one of great unease in the ecclesiastical landscape as traditional powers were in decline. It's possible that the bishopric of St Andrews began around this time, though that is merely the most widely-held scholarly speculation. If you want to delve into more nitty-gritty details of changes going on in the mid-late 9th century, read Alex Woolf's Pictland to Alba (Edinburgh, 2007). You'd also probably enjoy Thomas Charles-Edwards's The Chronicle of Ireland, which is his reconstruction of the different chronicles being kept in Ireland and Iona in the early middle ages. You could read through the entries around the year you're looking out to pick out notable names (abbots, bishops, and kings usually had their deaths recorded) and use those as a jumping-off point for further research.

I know more about Scotland than Ireland so my answer was slanted more towards that, but hopefully this helped! (I did my undergraduate thesis on the Pictish church around St Andrews from the 5th to 9th centuries.)