At last I've managed to connect the LimeSDR to an Arduino Due. I was delayed by trying to mess about with JTAG adapters and only solved the problem by making a custom PCB to breakout the 0.05 inch pitch JTAG connector:

The SPI connection has been tested and is working nicely. Now for some brain frazzling hex to binary register manipulation to load my repeater settings via the Arduino and then control it by the Arduino. Standalone repeater coming soooooooon!