Consider $A=BC$. A has orthonormal columns. I want to verify that the inverse $D=C^+ B^+ $ satisfies Penrose properties.
The following holds: $ CDC = ABB^+A^*AB = ABB^+B \overset{!}{=} AB = C $
Why can I state that $B^+ B= I$?
Source: https://en.wikipedia.org/wiki/Proofs_involving_the_Moore%E2%80%93Penrose_inverse
We want to show that if $C = AB$, $A$ has orthonormal columns, and $D = C^+B^+$, then we have $CDC = C$ (as would be required in order to have $D = C^+$).
As you have correctly shown, we have $$ CDC = ABB^+A^*AB = ABB^+B. $$ Although it does not necessarily hold that $BB^+ = I$, it does hold that $BB^+B = B$ (since this is one of the axioms). So, we indeed have $$ CDC = A(BB^+B) = AB = C, $$ as was desired.