You can use this website to see where both names were common in Ireland in the mid 1800s. Note the box for adding the second surname.
McWatters is fairly exclusively associated with Ulster (the northernmost province of Ireland), and the places where the 2 overlap and nearly all around Belfast and Counties Antrim & Down, in Ulster.
The problem you face is that for someone born around 1773 there are not many records in Ireland to search. Only a handful of churches have baptism and marriage records for that period, and of those that do, not all are on-line.
In the 1901 census all the McWatters in Ireland lived in and around Belfast. All were Protestant, either Church of Ireland or Presbyterian. So I’d guess your ancestors were Protestant too. The churches usually hold the original birth and marriage records but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
You will only find a small number of churches with records for 1773.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).