Get the FE written!
The cost of the FE written is a small price to pay for the chance to get an interview, and it is a minimum requirement. I don't think you will get an interview without it, so if you spent the $50 to apply just take another $250 to ALLATPs and get the written. Or just study the material and go take the test. In any event, you must plan on getting the FE written if you want to work here so you might as well bite the bullet now and show them that you are serious about the job.
Also, if you have the FE written and it is expired you will need to have it current when you start training. For some reason FedEx does not accept the exemption about the written being good indefinitely if you are employed by a Part 121 carrier. Several people have asked that specific question and during the new hire indoc training the instructors were clear about that. Don't know why, but that is the way it is.
Best of luck.
FJ