I dont know the anser to the specific question. It seems that for the study of hyperbolic Coxeter polytopes even if using some properties of general simple 4-polytope one needs to use the very restricted nature of these Coxeter polytopes. Overall the question of prescribing the sizes of 2-faces of simple (and general) 4-polytopes (and d-polytope) is very interesting. For $d>4$ it is known that there is always a 2-face with three or four edges and this is easy for the simple case. For 4-polytopes it is conjectured (by Igor Pak) that if all 2-faces has at least 5 sides then the number of $k$-faces is at least that of the 120-cell. (This is quite delicate as it does not extend to duals of triangulations of homology spheres.)

The following answer http://mathoverflow.net/a/24340/1532 to Characterizing faces of 3-dimensional polyhedra. (Related to Victor Eberhard's Theorem [1890]:) fives some related results and questions.

To attack this question we can try to use The Dehn Sommerville relations to study the sequence $n_3,n_4,\dots, $ and to estimate the average number of sides in a polygon. This number is below 6 and it can be expressed in terms of two parameters of the dual polytope $g_1(P^*)$ (essentially the number of vertices) and $g_2(P^*)$. Understanding the sequences $(n_3(P),n_4(P),\dots )$ for 4-polytopes (and even very special classes like stacked polytopes) and in particular the cases where $n_3=n_4=0$ is indeed very interesting.

I dont know the anser to the specific question. It seems that for the study of hyperbolic Coxeter polytopes even if using some properties of general simple 4-polytope one needs to use the very restricted nature of these Coxeter polytopes. Overall the question of prescribing the sizes of 2-faces of simple (and general) 4-polytopes (and d-polytope) is very interesting. For $d>4$ it is known that there is always a 2-face with three or four edges and this is easy for the simple case. For 4-polytopes it is conjectured (by Igor Pak) that if all 2-faces has at least 5 sides then the number of $k$-faces is at least that of the 120-cell. (This is quite delicate as it does not extend to duals of triangulations of homology spheres.)

The following answer https://mathoverflow.net/a/24340/1532 to Characterizing faces of 3-dimensional polyhedra. (Related to Victor Eberhard's Theorem [1890]:) fives some related results and questions.

To attack this question we can try to use The Dehn Sommerville relations to study the sequence $n_3,n_4,\dots, $ and to estimate the average number of sides in a polygon. This number is below 6 and it can be expressed in terms of two parameters of the dual polytope $g_1(P^*)$ (essentially the number of vertices) and $g_2(P^*)$. Understanding the sequences $(n_3(P),n_4(P),\dots )$ for 4-polytopes (and even very special classes like stacked polytopes) and in particular the cases where $n_3=n_4=0$ is indeed very interesting.