I can now program the Attiny 4313 with the Due if I have the Attiny in a breadboard. It only fails about 5% of the trials. If I put the Attiny into a socket embedded into my local SPI bus, programming fails about 95% of the trials.
It does not seem to matter if I select the HARDWARE_SPI option of ArduinoISPdue or bit-banging. The SPI clock divisor is set at 225, so I don't know that I can go slower. At this point, I am having good enough success on the breadboard, but SPI programming seems real touchy when I try to integrate it into my PCB. I have checked all the traces for continuity, resistance, etc. On the logic analyzer, I can see the Attiny responding. The SPI signal levels look correct, but I keep getting device signature errors. It is as though SPI is so marginal that the traces and connectors on the PCB throw it off. The same circuit with the Mega worked fine. I will continue to see if I can isolate the problem.