I see no reason why you couldnt bulk insert within a for each enumerator (please dont lynch me, 3 weeks of SSIS is hardly enough to deliver expert opinion).

I would have a variable called strFileName, which I would add to the variable mappings section of the enumerator

I would also have a variable which has the folder name to traverse on, which I would add under the expressions section (directory name). Simply to assist when porting the package elsewhere

Then I would add this same variable to the connectionmanager of the file, under the expressions (connectionstring).

This should enable you to pass the file from the enumerator to the connection manager each and every time.

Good Luck

~PD