There is none. The ends don't justify the means (Kantian ethics).
There are network effects of a giant for-profit organization exploiting the jobless, vs a non-profit that is regulated to be aligned with the jobless.
If you're using rational beings as a means to your ends, they need to be part of the ends, which is why Amazon is being criticized.
You're committing the fallacy described by the "Copenhagen interpretation of ethics". http://blog.jaibot.com/the-copenhagen-interpretation-of-ethi...
First, why do you describe offering someone a job as exploitation? It's totally not. Giving someone an option they didn't have before makes them strictly better off. They can choose between it and their other options. Working a job when you didn't have one before tends to make a person better off - either way it's up to them, and either way, it's one more option they didn't have before.
Second, just by interacting with the situation, you don't become responsible for it. The homeless' situation is not a potential employer's responsibility any more than it is yours personally or any other person's on the world. What are you doing about the homeless problem? Employers who go out of their way to offer homeless jobs, and to indicate that they won't discriminate based on that background, are actually giving them the opportunity to improve their lives. By comparison, most people involved in the thread will have done nothing at all to help homeless or give them opportunities, despite criticizing employers who do.
(Please read the link above; it offers a useful discussion of the issues.)
Is this always true? Imagine a third world country where children die from lack of food, clean water, and medical care. Now imagine a group that goes in and offers to exploit the children (I'll leave how up to your imagination) in exchange for meeting all their basic needs. Is this group making the children better off... and conversely, are anyone fighting for laws to ban the actions of this group working to make the children worse off? Almost every single person I've brought this up to will say yes for at least some forms of exploitation. If you go with child labor, you'll get a decent number on either side, but if you go with other forms of exploitation the rate who say that offering the choice is harmful jumps to basically 100%. (This is one of those questions that, if I ever make it rich, I would fund official research into, but until then I will admit this is based off of my person anecdotes.)
All that being said, I agree that you shouldn't be responsible for the underlying situation. Even if you are wrong for offering a given bad choice, you are not at fault for the underlying situation.
Which fallacy is Kant committing exactly?
Yes it is true that if a company helps but does not entirely solve a problem, they are not obligated to stick it out. But in this case it may be that they are not helping by offering homeless people seasonal jobs. The fallacy here is that homeless people have a choice: if I were in a desperate situation, then I would take any job on offer. I.e. by being given the option the decision has essentially been made for for me, and whilst I may be better off in the immediate future, that doesn't mean it will last (and it could even be worse in the long term), and whilst I could decline the offer, and wait for something more stable, my circumstances are sufficient motivation for me not to do that.
A cynical person might even posit that by explicitly not doing enough to bring people out of homelessness, Amazon succeeds in maintaining a pool of willing and cheap seasonal labour.