Like others have posted, at under 200 yards, just about any bullet from a .300 Weatherby put in the heart/lungs of a deer will easily kill it. Accuracy is the key, and only your rifle will show you which bullet it will shoot the best. Meat damage is another story. Usually heavier and/or stouter bullets will cause less meat damage than lighter or more fragile bullets.
I only hunt with reloads and I have only hunted with Barnes bullets in my .300 Weatherby. My .300 Bee likes Barnes bullets and I have worked up moa or better loads for it with 168 and 180 grain TSX and TTSX bullets.
I have never killed a deer with my .300 Bee, but I have killed a score of deer size critters with it on hunts in Texas, New Zealand, and Africa with 168 grain TSX and TTSX bullets. Most were one shot kills and unless the bullet hit a major bone, most bullets were complete pass throughs with very little meat damage.
I primarily bought my .300 Wby for an elk rifle, but so far I have only shot two bull elk with it. Both were 5x5 bulls, shot at under 200 yards, and one shot kills. I shot the first one when he was standing broadside to me and my 168 grain TSX went in behind one shoulder and exited behind his other shoulder. He took 3 steps and fell dead. I don't think I lost a pound of meat with that shot.
When I shot the second bull he was standing quartering me, facing and close to a fence on property that I didn't have permission to hunt. So I broke my own rule of not shooting an elk in the shoulder, and put my 168 grain TTSX bullet on the tip of his shoulder. Again it was a one shot kill and because he was on a steep hill, he rolled farther than he ran. That bullet hit his shoulder bone just below it's junction with his shoulder blade, went almost completely through him and stopped just under his skin on his opposite ham. At least half of his shoulder meat was bloodshot ruined.