Responding to the actual OP, I must say first that ‘fairness’ is itself a notion even harder to parse than property. We have multiple, sometimes contradictory notions of fairness and we have to be careful when using it.
I think that fairness would completely undermine a view of property like Locke’s. If we want to take fairness into consideration, being the person that mixed their labour with previously unowned resources is irrelevant to who ought to acquire the new resources. It would go to the ones such that the situation would be the fairest, presumably something like those with the least resources.
I will add that Locke’s theory of property is also based on a different kind of fairness. I don’t think saying that it’s fair for the person who worked hard to obtain something should obtain that very thing is misusing the word ‘fair’. So, we get to the idea that ‘fairness’ is many things that don’t necessarily bode well.
To me, this suggests that we have different values, in particular here, different notions of fairness that we like but we have to, perhaps, prioritize some values over others when they clash if we want a coherent moral theory. And you have to hope that this theory is the “correct” one.
I personally am more inclined towards property à la Locke than something based around whoever needs it most.