It's likely (absent better data) that the problem here is that the CM and CM3 have "only" 4GB of eMMC flash on them. That isn't enough for a full Raspbian install.
There are a number of solutions... The simplest is to switch to the NEC "CM3-16", which is a custom, but generally available CM3 with 16GB of eMMC flash on it. Other solutions will require a carrier board set up for this situation. As much eMMC flash as desired can be put on the carrier board and a CM3L will boot from that. Alternatively, an SD slot can be put on the carrier and used as a boot device for a CM3L. Other alternatives are to set the boot from USB OTP bit and boot over the USB bus from ones choice of device. This would also work with a CM3 if the on-board eMMC flash doesn't have a bootable image.
And, by the bye, there is an entire forum here devoted to Compute Module issues. Detailed advice can probably be had by moving this thread there.