LTC7862 can't be started with load, but it can be loaded after no-load start.The schematic is as follows（The schematic is a reference chip manual）
I noticed that R6b resistor that is connected to VIN pin of the LTC7862, its value is too large. I would suggest using 2.2 ohm with the case size of 1206 instead of 2.2Kohm. See schematic attached with R6b resistor indicated by a red circle. This is the cause of the startup problem since VIN can't supply enough current to DRVCC pin during soft-start switching mode.
I also noticed that DRVUV pin must be connected to INTVCC pin with a zero ohm resistor with the FETs that you used for this design. Do NOT populate and remove the zero ohm resistor from DRVUV pin to GND.