In programming, it's common to write a class that both reads and writes files. What do you call this type of thing? Is it a file handler? A file parser? A file provider? A file streamer? How about a filer? Or do you just call it a file reader and leave it at that?
Your first response is probably the best, along with file processor.
processor: a machine that processes something.
process: a series of actions or steps taken in order to achieve a particular end.
A parser implies that it only reads the files, not writes them.
A provider implies it is the source of the file, not an intermediary.
A streamer implies the files are either being written continually or converted continually.
A filer implies that the files are stored without modification.
A reader implies the files are read without modification.