Sketchy data, but here's a guess. 3000' of 13.8 kV cable with shields bonded at both ends and tied to ground will tie the remote utility system and the local grounding system together.
Assume the circuits that are tripping all have ground fault protection and have some anomaly in the way they are wired - poor neutral to ground connection, poor grounding connection, multiple neutral to ground connections, ground wire through GF CT, etc. (Heat trace circuits have GF per code. Many 480V heating elements also have GF. Don't know how your 277V lighting gets in this category).
Assume that the utiilty experiences a 13.8 kV fault somewhere else on their system and the return fault current flows in the lower impedance path of the cable shields and the new facility ground system creating ground currents in unexpected paths. The ground currents flow through the GF CT's causing breaker trips.
This did happen at one of my old facilities. No faults occurred in our plant but a nearby utility line fault current liked our copper paths a lot more than the earth return. The currents tripped multiple miswired GF systems that had operated OK for years.