Given the problems I've had trying to configure in slave mode using the RPi, I decided to see if the FPGA would configure itself in master mode. So I removed the CAT Board from the RPi and made connections to the power, ground, SPI I/O, and RESET like this:

The serial flash has been removed so there's no way the FPGA can actually configure itself, but it should at least try to read in a bitstream. But toggling the RESET or power supply didn't produce any activity on the SPI clock line. It's possible the FPGA is completely dead.

My plan now is to place an FPGA on another CAT PCB with a minimal amount of extra circuitry (mainly the voltage regulators) and see if that shows any signs of life.