How to address and disable the ASIX Ethernet controller extended low-power sleep mode on the Rabbit RabbitCore RCM43xx?

You should be able to adapt Samples/BL4S1xx/lower power.c for use on the RCM43xx. It shows how to use ifdown() and pd_powerdown() to turn off IF_ETH0, and then use pd_powerup() and ifup() to bring it back up after going into sleep mode.