I am now calling this thread [SOLVED] because after reasonable investigation (using Devel::Cycle and Test::Memory::Cycle and other tools suggested), I conclude that the program is ... pure and simply ... too big for the narrow confines of a “shared hosting” deployment, and through no fault or error of its own. The site will be deployed from the outset using Managed or Dedicated Hosting.

From the start, my test-suites for the so-called Context object structure, which is what everything else related to a request “hangs from,” already check that there are no circular references (that are not properly “weakened”). So, I was pretty confident of that.