OK, here's what I'm doing distilled to only the System.Management calls:
Simple queries and Method invokes work over the same connection. This query won't. And the file exists on the remote machine. Clues?
myQuery = "Select * from CIM_DataFile Where Drive = 'C:' AND Path = '\\Users\\someguy\\Documents\\' AND FileName = 'Default' AND Extension = 'rdp'";
options = new ConnectionOptions();
options.Username = myUsername;
options.Password = myPassword;
options.Authority = "ntlmdomain:MYDOMAIN";
scope = new ManagementScope("\\\\REMOTEMACHINE\\root\\CIMV2", options);
scope.Connect();
searcher = new ManagementObjectSearcher(scope, new ObjectQuery(myQuery));
myResults = searcher.Get();
ManagementObjectSearcher.Get() gets me a ManagementException saying "Invalid query." A simpler query, like, say, "SELECT * FROM Win32_NetworkAdapter", works.
I tried to reduce the WHERE's to just one, i.e. "Select * from CIM_DataFile Where Extension = 'rdp'". It works, although obviously it doesn't get me what I want. (Before the edit I mistakenly thought it didn't work even then; see comments) I'm at the end of the rope here.