Without having seen the actual verbiage of the communication you had with them, "fee approved" sounds like a go-ahead to me.  If they didn't want you to start the work, they should just ask for the ETA first instead of saying anything that would indicate that you were being granted approval to begin work.  Having been on the client side of this sort of thing, of course they should pay.  It was their own sloppy wording that created the issue and in the end they got what they wanted.

I hope you can make them realize that this is their own fault and get your money, but if not just report them here and elsewhere and never work for them again.