So I decided to get technical here, and started to read through the Lisa boot ROM code, available here: http://www.1000bit.it/support/manuali/apple/lisa/lisaBOOTROMLISTING.pdf
It seems that the point where the testing icons come on the screen is at memory position 0B96. My Lisa doesn't even get to this point, so I imagine my repeated clicking error happens somewhere before this point in the ROM. I'll keep looking.
Just thought I'd post my findings in case anyone else wants to dig through the ROMs and track this problem down.