I am not sure whether I really understand what the author argues for, probably because I read the paper in a rush. Does he argue that [the choice of] a logic is in some sense equivalent to [a choice of] what kind of entities exist?

If that is indeed the case, I would tend to disagree. While we often think of logic as a provider of undisputable truth, I am not sure that this is justified. There is more than one type of logic and they lead to different conclusions or contradict each other in the general case. Therefore it seems to me that the obvious thing to say is that logic is a human construct we use to reason about the world, it does not reflect universal truths but was invented and constructed in a way to be useful to deal with our world.

In consequence it is obviously very likely that logic and metaphysics are strongly correlated, but that would hardly be any deep insight but just a consequence of the way we constructed or choose the logic we use. If the logic I use to deal with the world makes a prediction about the world, for example the existence of a certain entity, then that entity might actually exist if may logic is a good match for the world, but I see no reason why the entity would have to exists, my logic could just be a bad choice.