Unfortunately you need access to the Edison's USB port in order to flash your board in any of the supported ways of flashing your board.
The only way (that I can think of) you would be able to flash your board without access to Edison's USB port is to have flashed the Edison with the latest image before putting it on your custom board so the re-partitioning would be already done and then storing the image in the Edison's flash memory to reflash it with the old method of reboot ota.
But this is not compatible with the 159.devkit image since it is sized more than space available on the Edison's flash memory therefore this image can only be installed through Flash Tool Lite or flashall.sh. But the 159 image does fit the Edison's flash memory and if the method I described works for you then it might be your best option.
We now do this: image to be loaded into the Intel Edison Module in Arduion Board, and then move the module in our product.