Only "stronger" works in the blank.
I guess your confusion comes from our use of "(verb)ing" in the place of an adjective sometimes. Well, the first issue is that "become" indicates a change, so the comparative "stronger" works. "Become strengthening", or any (verb)ing for that matter, just sounds awkward.
The other issue is, "strengthening what?" If you were to put "strengthening", you would have to rewrite the whole sentence.