Old Wolf is correct, running a device for a longer period of time will not increase current draw (i.e. current will not become greater than 5A), this is because Ohm's law states I=VR (I=current, V=voltage, R=Resistance) and the voltage and resistance cannot change on this circuit.
Inrush current is a condition associated with transformers (your halogen light circuit) or electric motors mainly, not horns. It is caused by the resistance of the windings being low at start-up because of no inductance (transformer) or back emf (motor) being produced, so again back to Ohms law I=VR when resistance is lower the current draw increases.
My vote is for no relay, for two reasons: there is a fuse in line with the horn, and less wiring to complicate things and break down!
