Limey, nice try but it does not work that way. Any training for a FI rating would be for either FAA or JAA, not both at the same time.
As for doing FAA first then adding JAA, when I did my CFI rating (conversion from CAA/509 approved instructor) I remember reading in the regs that there was no formal course of training for a CFI, but maybe thats changed now. Given that scenario, you would have to do the JAA first.
Upshot is, do one then the other, which way round would have to be worked out by someone who knows and trains to both systems.
Bramsen, ask the manager you spoke to to explain the logistics of it all, then tell us!
To answer your question, I think he is right that you would be more employable.