This is not exactly what you seek, but at least you can find educational research articles
under the key phrase dimensional analysis. Below  says it's useful,  questions that usefulness.
 Hrayr Ohanyan. "The Application of the Method of Dimensional Analysis When Solving Problems."
American Journal of Educational Research.
Vol. 4, No. 1, 2016,...
This is pure speculation — I don't have enough information to know what the student was thinking — but they may have only known how to compute the perimeter in one way: taking the lengths of each edge and adding them up. (They may have even unconsciously identified the perimeter as exclusively being the result of that procedure.) However, in the given figure,...
difficulty in understanding the interrelation of mathematical constructs.
difficulty of fitting a mathematical problem on just one line
difficulty in compressing the problem to a communicable slogan
Give her a collection of 5 such shapes on graph paper, all of which have the same perimeter, and have her compute. Then see if you can talk about why these shapes all have the same perimeter. See if she can draw another shape on her own which has the same perimeter. Make sure that all of these use "the same trick".
Your approach sounds a bit stiff to me. Put a picture inside the shape. "Adding up the lengths of the edges" might be "How far is it round the edge?" for a start. It doesn't take long before the picture can be dropped and better mathematical language can be introduced, unless your pupil has limited learning skills.
"Adding up the lengths of the edges" really is an adequate definition for the perimeter of a figure. The concept she might be missing is that it doesn't matter how long the individual edges are, the perimeter is always the same.
I would try explaining this to her using a challenge. Give her some grid paper (if she's used to it), tell her the length of all ...
I think the OP asks a difficult question that will not have a succinct answer.
Permit me to point to one publication,
Schoenfeld, Alan H., ed. Assessing Mathematical Proficiency. Mathematical Sciences Research Institute Publications. 53. Cambridge University Press, 2007. (PDF download of book.)
which addresses the question of what constitutes mathematical ...
I often see a "shift" among my fourth graders when we move into geometry units. The students who have been quite successful with computation and even algebra concepts begin to struggle to "see" things in geometry...similar to what you describe. What's helped is putting the area and perimeter piece on hold until we've done a thorough investigation of the ...