I would start with sending the CASA form to the CASA and see what happens next.
You might want to fill in the FAA form already, make a copy and send that as an appendix to the form you send to the CASA, together with a copy of your license, medical and everything else that might be relevant. If the CASA doesn't need it, they can always discard or disregard it.
But I would not start with sending stuff to the FAA. After all, on the FAA website it states: "airmen from the United Kingdom or Australia must contact their respective CAA to complete additional forms that are required PRIOR to providing the requested information to the Airmen Certification Branch." So without the CASA letter of validation, the FAA can't do anything.