There is a canonical thing to do if you have a lattice quotient of weak order on the symmetric group.  Namely, you take the Coxeter fan (which has one maximal cone for each element of $S_n$) and you glue together the chambers that correspond to the same element in your quotient.  This gives you a fan.  The natural thing to hope for is that this fan will be the outer normal fan of a polytope.  Checking whether or not that is true, is somewhat delicate.  A thing which sometimes works is to consider taking the permutohedron and removing some of its defining inequalities (so that the resulting polytope gets bigger).  It is possible to construct an associahedron in this way.  

However, I checked, and the map you have described is not a lattice quotient of weak order.  So I do not know of any canonical thing to do. 

However, if I have correctly understood your definition of $YT$, the number of 0-cells in $YT_n$ is $2(n-2)\choose n-2$.  This is the type $B/C$ Catalan number, and suggests that the thing you want to construct is the cyclohedron (also known as the Bott-Taubes polytope).  Its vertices are naturally in bijection with $YT_n$.  It isn't naturally constructed starting from the type $A$ permutohedron, but rather from the type $B/C$ permutohedron.  I can add more details if this seems like it would be useful.