Not sure what happened, the message went as soon as I started to type, the regiment you are talking about is presumably the one described in the articles on this website as the 'North Carolina Highlanders'. Not too sure where the 'Royal' tag fits in but it seems that the regiments exact name was flexible, Lt Donald MacCrummen, one of the last of the Skye piping family who served in the regiment describes it in his own words simply as ' the Highland Militia raised in North Carolina in 1774'. He then moved on to the 84th in 1776, for which regiment he 'procured' 42 men, then in 1778 he was removed to the British Legion before being captured just after the battle of Camden.

As far as the names go the only one included in Blacks 'The Surnames of Scotland' is Hyndman, which he describes as being of Renfrewshire origin, although he does note an 'Annie Heyman' who was at a meeting of witches in Bute in 1662 and suggests that her name was intended for Hyndman. The rest of the versions you give all seem to locate in the far south of England, however returning to the Hayman version of the name, I remember seeing it among some Kintyre names in the early 18th century although it no longer exists there today. If your ancestor was indeed a Gaelic speaker then Kintyre was a mixed Gaelic and Lowland population by the 1700's and there was some emigration to Carolina I believe, so might be worth investigating further.