Creekrat72, I don't want to come across as a jerk but if you're letting a 12 year old kid shoot a lightweight 300 Winchester, you're setting him up for failure. Bad shooting habits are hard to break, especially for a kid, and putting him behind a rifle like that is sure to create at least some of those habits. If he can't shoot a 243 well maybe he's not ready for big game hunting??
The 308 Winchester is plenty of gun for deer, bear, and elk at reasonable distances. Pick a good bullet and be able to place that bullet in the kill zone and it'll kill just as quick as a 300 Winchester.