On the 22R-E there's a bypass in the diagnostic plug that forces the fuel pump to run with the key, eliminating the interlock in the AFM. If that makes it work then you are probably not dealing with a fuel pump issue. If it still does not start, then I would check the relay next. If neither of those is the problem, then it's got to be something making the ECU unhappy, sensor, broken wire, clog in the fuel lines or something. But you'd have to look in the FSM to find the connection to make the fuel pump run with the key. Also being a 1FZ the EFI system is more complex than a 22R-E or 3FE, so the conditions to make the ECU happy are somewhat more touchy. Oh, also pull the codes, that might tell you whats wrong. I assume the 1FZ does the flash-the-CEL trick.
"Silent enim leges inter arma." -- Cicero