Your car needs a fuel pump either way. Trust me.
The 5V at the pump is a supervisory Voltage the DME sends to the pump.
12 Volts or something close to 12 Volts is only supplied for the "Prime" cycle which is for only 2-3 seconds when the key is initially turned on. Many Voltmeters are not able to capture the prime cycle.
You may be able to capture 12 Volts while cranking, however, depending on how the circuit is wired the relay may not pull down without a load? I cannot recall the exact behavior of the circuit for sure.
This happens ALL the time, where there is too much focus on measuring 12 Volts at the pump and it cannot be reliably found!
If you want to swap the horn and fuel pump relay, they are behind the glove box, door and liner need to be removed, they are the darker green relays that are the same color. I think the far left and middle relays?