Even though logic runs on 3,3V normally LCD analog signals are differential with analog bias voltages in the range of 12V. It seems these are the ones getting interference, not the logic or SPI interface.
Consider interface shielded wires, ferrite choke sleeve, ground plane, check all analog signals and voltages and ensure Analog ground is clean relative to analog driver. Imbalance from stray EMI can induce noise.
But scoping analog signals on LCD will show the problem, or show us.