Abelianization is a functor, so yes it is obvious the face and degeneracy maps extend.

The standard example of a non-trivial loop in the space of generalized Morse functions which is no longer a loop after adding framings is the swallowtail move. This starts with a single index i critical point, then introduces a canceling i and (i-1) pair of crit. points, and finally cancels the original i critical point and the new (i-1), leaving a single index i critical point behind. It is show in Figure B in section 1 of Igusa's paper "The Space of Framed Functions".

Does the 3-sphere admit a cubulation in your sense?