“What is the best cartridge to hunt mule deer out to 400 yards?”
Your first question should NOT be, “What’s a good 400 yard Mule deer cartridge?” Your first question should be, “Am I a good 400 yard shooter?”
Take whatever rifle you fancy to your range and set up a big game target at 400 yards at a broadside target angle. Start from standing, drop to any shooting position you fancy (hopefully prone with sling, but not benchrest) and fire one shot, untimed. Repeat for a total of 5 shots. Now go down range.
This test simulates perfect hunting conditions. There is no time pressure (i.e., a target on the range won’t move) and you are given a gift with a clear, broadside shot. All 5 shots should be centered in and around the vital zone in an approximate 12 inch group (depending on how big the animal/target is.)
Assuming you check out, repeat the same drill, but time each shot. If you can accomplish this again while timing yourself, regardless of what the timer says, I would say you are qualified at 400 yards. Cartridge choice is academic, as nearly any suitable centerfire for hunting will suffice.
If you can’t accomplish this, don’t have the range to try, or worse, won’t take the time to try, your question should be, “How do I stalk closer if I spot an animal at 400 yards?” Don’t attempt the shot at that distance as you have no idea if you can pull it off. The cartridge your rifle fires won’t make a difference.