Here are a few realities which have been completely ignored in the debate of physics above:
1) Velocity is a fundamental parameter of Kinematics. Mass, Time, & Velocity are the foundation of all other parameters. Relativisms among these three parameters are the purpose for most other calculated values among the entire study of physics.
2) Kinetic Energy is NOT conserved in real world, inelastic collisions - MOMENTUM, however, IS conserved. Which is WHY we (using "we" to refer to myself and other engineers and physicists which actually do these calculations) use MOMENTUM to calculate the WORK done within the system - Work being defined as the difference in Energy before vs. after. So we start with a given mass & velocity, which is used to calculate both the kinetic energy and momentum of the projectile. A common misconception is that if a bullet stops within the animal, its entire Kinetic Energy is "dumped" into the animal, because it isn't - we know we have Kinetic Energy lost as heat, sound, and destruction to the bullet itself. (Just for fun, the Rate of Work, is Power - Work/time - meaning how fast is the Work being done... But this is differential calculus in the case of a decelerating bullet, so it gets pretty messy pretty quickly).
3) No, the TKO is NOT a relevant value for any other application in any other branch of physics or engineering - but it IS simply the product of momentum and bullet diameter. In this case, it has been used with relatively high success as a proxy for understanding momentum transfer - simply, a bigger diameter bullet hits more animal than a smaller bullet, so the same momentum on a bigger diameter bullet will transfer momentum more rapidly (which means transferring KE more rapidly, aka, doing more WORK in less time, impacting with more Power). Sectional Density is another proxy for this same consideration - it's not so different than considering the Ballistic Coefficient of a bullet in flight as a proxy for its Coefficient of Drag (not so different than things like "slump factor" for mixing concrete either - an indirect measurement used as a proxy for a relatively complex and relatively unmeasurable quantity that is Work(done to animal) and KE(out). So we can look at something simple like bullet diameter or sectional density as proxies for for the more complex value. It's not "good," but it's better than nothing, and it's certainly closer than KE since all of the terms in momentum, and subsequently TKO are linear rather than Power Functions.
So it does remain fair to point out that Kinetic Energy is an exceptionally poor proxy for the killing potential of a bullet - but it's also impossible to have high momentum without also having significant kinetic energy. We can't REALLY only use TKO either, but it's a hell of a lot more relevant to how the rest of physics is done than is a simple statement of KE.
SO.... What really remains, is the fact there's no definitive science either way which suggests 338wm is a better or worse choice than 300wm for the OP's application...