I could be wrong on this so go easy everyone.......
There is no physical thing that is an fATPL, it is a term used to encompass the CPL/IR, which obviously matures to an ATPL at the required experience - which is an actual licence.
I would have thought sending them both together would perhaps be cheaper - my school told me to apply for multi at the same time as CPL as they would put the rating on for "free" - again check this.
Either way if my PPL issue was anything to go by you probably won't have the CPL in your hand in 3 weeks to send off again. Think I have just said I'd do it together...........I think