Maybe the pyramid over a regular polygon would be a good example.

And for $E_7$, one gets $221714415 = 3^6 * 5 * 13 * 4679$.

There seems to be some divisibility by h/2, where h is the Coxeter number. The Auslander-Reiten translation acts on the set of bases for the K-group. The order of every orbit divides h. Not clear to me what the stabilizers can be.

For $E_6$, one gets $846720 = 2^7 * 3^3 * 5 * 7^2$.

For $D_6$, one gets 228055 = 5 * 17 * 2683. This does not match the number of maximal chains in the noncrossing partition lattices.

Maybe use the level of the unique inner vertex under the rightmost leaf of the leveled tree to fix the painting height ?

There are small trees with wild representation type, for instance trees having one more vertex than affine Dynkin diagrams of type E.

There is a necessary condition, that the Coxeter polynomial is a tensor product of two polynomials. This can be checked for instance on the set of roots.

In singularity theory, one can make the simple singularity E6 ($x^3+y^4$) from A2 ($x^3$) and A3 ($y^4$). This sum-with-disjoint-variables is named the Thom–Sebastiani sum.

Can this be translated into an Apery-like sequence ?

Is the power $t^{2n}$ before $P_n$ correct ? Maybe it is rather $t^{n}$ ?

Link for Cardy article : doi.org/10.1016/0550-3213(91)90024-R

For the next permutohedron, the squares are no longer facets, so one cannot just remove the squares by removing the associated inequalities defining the polytope, as was the case in dimension 3.

What ? See en.wikipedia.org/wiki/Primitive_element_theorem