Note: This was originally posted by an inactive account. Content was preserved by moving under an admin account.
When you say you are using pipe delimited txt files, am I right in presuming you are writing out the data using Output CSV/Delimited node? If so there is a Field Order section in that node where you can rearrange the order of the fields and also you can remove any fields that you don't want. In this case if FileName on the input only serves the purpose of telling the node what the output file should be called you could remove it in this section and it would not be on your output.
Another option is to write out the file to some generic file name (with FileName removed from the fields), then using a transform node directly after you have written out the file (clock it to your output node), you would use os.move(GenericFile,NewFileName) command.