The 5k LG monitor uses DisplayPort MST to drive the monitor, which is supported on the MBP on a single cable. However, when you use the eGPU, the GPU card drives the external monitor, which is standard DisplayPort. The TB3 ports are used to extend the PCIe interface from the MBP to the eGPU. It's designed to communicate with the eGPU or daisy-chained eGPU's. it's not designed to drive a monitor.
The only way it would work is if the LG 5k supports dual DisplayPort inputs from the GPU.