So it turns out the SvnStatusArgs class has a "RetrieveAllEntries" boolean flag that defaults to false.
As the name implies, setting this true returns every file, whether it was modified / unversioned or up to date.
1 extra line in the CheckSVN() method in my original post:
SvnClient client = new SvnClient();
SvnStatusArgs sa = new SvnStatusArgs();
sa.Depth = SvnDepth.Infinity;
sa.RetrieveAllEntries = true; //the new line
Collection<SvnStatusEventArgs> statuses;
client.GetStatus(path, sa, out statuses);