Compare the ampere rating on the original horn vs. the air pump. It may be the OEM relay and circuit isn't rated for what the air pump draws. There are relays available for air horns if you search the web. This one is rated at 80amps per the picture.
https://www.ebay.com/itm/12V-Universal-Dual-Tone-Electric-Air-Horn-Wiring-Harness-Relay-For-Car-Truck-Van-/312679248735?_trksid=p2385738.m4383.l4275.c10