The broad historical background to the Reivers arrival in Ireland is that when King James I became King of England in 1603, he was already King of Scotland and so then became the first King of both countries. For hundreds of years the Scottish Borders had been fairly lawless and travellers were routinely robbed, and cattle often stolen and herded across the border by moonlight. The new King was particularly keen to stamp this out because he saw it as an obstacle to commerce between the 2 countries, and being joint ruler that bothered him more than his predecessors.
At the same time he had the problem of Ireland. The Spanish Armada had recently attempted to invade England and further invasions by the Spanish or French were feared. Ireland was seen as a possible jumping off point for such an invasion and understandably, the native Irish could not be relied on to support the English or resist any invasion. So the solution was to plant trusted settlers from England, Wales & Scotland in Ireland, in large numbers, to subdue the native Irish and be on hand to deal with any invasion. King James I was a Scot and so particularly favoured his fellow countrymen. Much of Ulster, including Tyrone & Armagh was heavily settled by Scots. During the 1600s, some 100,000 Scots settled in Ireland representing something like 10% of the entire Scottish nation. They didn’t all come as part of the Plantation. Some settled in the 1640s when General Munro's 10,000 strong Scottish army was disbanded at Carrickfergus after the 1641 uprising, and a further batch came in the 1690s due to famine in Scotland.
As far as the Reivers were concerned, King James I decided to move large numbers of them to Ireland. He needed settlers in Ireland and he wanted to get rid of the Reivers from the Borders, or at least stop the criminality by breaking their control of that area. So moving them to Ireland was a bit of a masterstroke which killed two birds with one stone. So if your ancestors are Reivers, they probably arrived in the first 20 years of the 1600s, as part of the Plantation of Ireland.
There are no records of individual settlers at that time. We do know the names of the big landowners (Scots & English) but not of their tenants and others who followed them. DNA might get you back to Scotland but there are no comprehensive paper records in Scotland or Ireland which record individuals names or arrivals.
If you visit Border towns like Selkirk, Hawick, Galashiels & Jedburgh, you will find a strong Reiver tradition with folk & food festivals every summer. You can go for a Reiver walk by moonlight and have a 16th century banquet. The usual nonsense. For example, see:
http://www.hawickreivers.com
There are a fair number of Mehargs in Ulster, and the common explanation for their surname is that it is Graham (a common Borders name) reversed. They were reportedly Reivers who objected to being moved to Ireland and reversed the spelling of their name in protest. How accurate an explanation that is, is debatable but it is widely believed.
Finally, regarding Methodism. (You mention that some ancestors were Methodist). Methodism took a lot longer to become established in Ireland as a separate denomination than in England. In Ireland there was considerable resistance to separating from the Church of Ireland. It was 1815 before Methodists agreed to conduct their own baptisms. However because of continuing loyalty and other factors, many continued to use the Church of Ireland for sacraments for many years after this date and it was 1871 before all Methodists routinely performed their own baptisms.
For marriages, the earliest ceremonies conducted by a Methodist Minister in Ireland that I am aware of, date from 1835 (Belfast Donegall Square, the first Methodist church in Ireland). However in the mid 1800s there were only a few Methodist Ministers in Ireland (Methodism relied heavily on lay preachers). So the shortage of Ministers contributed to the continuing practice of marrying in the Church of Ireland. In addition, in the early years, many Methodist Meeting Houses were not licensed for marriages so that too contributed to couples marrying in the Church of Ireland.
So to summarise, you are unlikely to find many Methodist baptisms before 1830. Few marriages before the 1840s and only a handful for many years after that. If there are no Methodist records in the location you are interested in, I would search the Church of Ireland instead, as that’s the most likely place to find the relevant event.
Not many Methodist Meeting Houses have graveyards and so they may be buried in public or Church of Ireland graveyards (which are open to all denominations).
So if your Irish ancestors were Methodist, then originally they would have been Church of Ireland. And that fits with Reivers. For some reason – and I have not been able to get a definitive explanation – Reivers and others from the Scottish borders tended not to be Presbyterian (the main Scottish denomination) and in Ireland generally appear to have joined the Church of Ireland.
Bear in mind that Beattie/Beatty is a fairly common name. In the 1901 census of Scotland there were 5800 and in Ireland 3767.