It's been my experience that accuracy and bullet placement is more important than bullet diameter and construction, especially for deer. I killed most of my mule deer with standard cup and core bullets, with the majority killed with 117 grain Sierra Spitzers from my .257 Ackley. I killed my largest mule deer, a 30" nontypical, with a .45 caliber patched round lead ball on a Colorado muzzleloader hunt.
From 1980 through 2003, I killed 21 elk with 180 grain Nosler Partition bullets at approximately .300 Win velocities. I felt that all of those bullets did an excellent job.
In the past 15 years, I've used Barnes TSX and TTSX bullets on hunts on 4 continents for critters as small as 25 pound Steenbok and Klipspringer to 1500 plus pound Cape Buffalo. Most were one shot kills, and I have NEVER had a Barnes bullet fail to do what it was supposed to do.
In the past 20 years I have been on 15 guided hunts and have hunted with 21 different guides. Only one of those guides was a handloader, and most knew very little about different bullets and ballistics. Most of what they knew about bullets and ballistics was from clients they had guided, and most of those clients also were not handloaders, and many were only occasional shooters. As a result many clients wound and loose game animals then blame their bullets instead of their poor bullet placement, especially if the bullets are a new design like the Barnes. So this is what the guides see, then they pass that belief on to their future clients.
So for a mule deer hunt with a .300 Win mag, should you use Barnes TSX/TTSX bullets or Nosler Partitions? I would say to use the one that is most accurate in your rifle, and enjoy your hunt.