You may not have to
It probably won't matter which concept you use due to the theorem of Ehresmann. See: http://en.wikipedia.org/wiki/Ehresmann%27s_theorem
It states something like most surjective submersions are in fact fibre bundles (most meaning that this is the case if the surjective submersion is proper, and I am not sure how dense proper maps are). Is there an approximation theorem for proper maps?
So i think the answer is that you don't have to. Also, (smooth?) fibrant replacement can be done to any map so that you get a LES in homtopy homotopy (although this map may no longer be a submersion.).
hope this helps, sean