I think your problem is with the NIC not with the card reader. The card reader is just a USB device or should be, which will be listed in lsusb that is how all mine are setup.
If you look up your list there you will see eth0 link down. I am not sure if this is because you are not far enough along in the boot process or you do not have a cable connected. You may have to install the driver at boot for that nic card. I am not 100% sure how to do that. but once you generate you image on the core you should be able to get into the diskless directory and install the module for that nic. Check realtek site to see if they have a driver for it yet.