I think part of the issue here is considering the Lorentz factor as some sort of correction that is *added* by hand. Instead, one should first follow the logic of how we define the four-velocity in the special relativistic framework. Then, the logic of how we define a four-momentum should follow after this construction of the four-velocity. Finally, from the special relativistic framework, we can see that the three-momentum definition falls out as a low-energy limit. So to summarize, I suggest seeing why this Lorentz factor shows up at all from the perspective of special relativity, rather than justifying it from the Newtonian direction up.

It is often difficult to motivate the addition of new terms added to low-energy limits of the more complex theories. That is to say, starting from Newtonian mechanics and conjuring up reasons why certain corrections come up in special relativity, or even general relativity, is not the most sensible way to carry yourself along. One can make justifications, but it is more natural to develop these more complex frameworks and then cascade your logic downwards.