[1]:https://math.stackexchange.com/questions/184817/if-f-x-simeq-y-then-x-cup-varphi-mathbbbk-simeq-y-cup-f-circ

I think  this can be  treated using a gluing theorem for homotopy equivalences which is discussed in my answer to this [stackexchange question][1]. You have to represent the realisation of a simplicial space as a repeated adjunction space, but that is not too hard.