This can be done with a method similar to the one I used in the prequel question.
The numbers are displayed below in base 13 to keep the square-ness. (A for 10, B for 11, C for 12 and D for 13)
Rotations, reflections, and permuting rows would give other solutions.
Observation 1 (trivial):
Observation 2 (non-rigorous):
There is a structure that precisely realizes these needs:
What is missing for optimality ( or: the dangers of intuition) ?
Here's 2 fundamentally different solutions:
I suspect there are many because these are literally the first two things I tried as I was playing around and in both cases I was able to just shove the pieces in and get a solution. I did it in the numerical order shown.