There has now been partial success. After switching solvers from spooles to pastix, I was able to get the unit tests to converge. Given the results from this thread and that, in general, cohesive-zone model calculations are inherently non-linear, I wondered whether or not I would need to switch to a better numerical solver.
What I was not expecting, however, is that the pastix solver only solves the unit tests when valgrind is analyzing the calculation. Obviously, it is unreasonable to have calculations slowed down orders of magnitude because “debug” mode is required and memory leaks are being tracked. Therefore, unfortunately, the user element is not yet production ready, and I need to continue digging more into the code to see what might be going on within my user element/ccx to cause these problems.