You need to put in a double throw relay. Disconnect the horn from the original switch entirely and connect it to one of the relay throws, connect your circuit to the other.

Or just use a single relay and put the horn and your other load in parallel as long as they both run on 12V.

The horn button gets connected to the coil of the relay.

Relays don’t have resistors necessarily in their coils, but the coil is designed to draw a certain amount of current at a certain voltage. In essence it is a resistor. If it were to pass the full current required to drive a car horn through the coil, it would catch fire.