You should use 180 class bullets or bigger with the 300 RUM. Why? Because you have a ton of powder, take advantage of it and produce some energy. Also because after several hundred yards, even though the MV is slightly slower it will still yeild a flatter trajectory to 1,000 yards due to a higher BC with SOME 180+ bullets. Also at 300 RUM velocities anything less might not hold up good enough to deliver damage far enough into a big critter. You need more mass to get the bullet to the vitals. These are just a FEW reasons.
Also I have had great luck with the 178 AMAX, 180 AB and the 200 AB. I use the 178 for deer, sheep and black bears. The 180's and 200's for everything bigger. It depends on my mood.
Long range shooting is a process that ends with a result. Once you start to focus on the result (how bad your last shot was, how big the group is going to be, what your buck will score, what your match score is, what place you are in...) then you loose the capacity to focus on the process.