Good point about the relay. OP would have an easier job if they operated the horn via a relay rather than directly from the switch. Radio Shack has (or at least used to have) an inexpensive 30 amp 12 volt relay, 30 ohm field winding as I recall. I installed one in my Corolla so that the ignition switch wouldn’t have to handle the starter solenoid current during cranking. Most momentary switches will be able to handle 30 ohms, which would be a current of only 12/30 = 400 mA.
Edit: Now that I look at @TwinTurbo 's link above, it appears that horn kit includes a relay. Maybe it is just bad luck, the kit had a faulty horn button-switch from the get-go.