I have used the 303 British extensively in the past and I see absolutely no issues with this bullet at all. My own preference was for the 150grain bullet and to be honest any shots I took with it were not outside the 150 metre mark.
At 300 metres the 303 still has very capable killing speed and punch although at this distance you are looking at about a 15 inch bullet drop if zeroed at 100metres.
As long as you can confidently account for the bullet drop on this calibre you should be fine.
The old Government Deer Cullers of the 50's, 60's & 70's in New Zealand were given ex military & surplus Lee Enfields in 303 for the purpose of culling out the prolific Red Deer herds of that era in New Zealand. As many as 2 million Red Deer were taken by the 303 at this time so it's attributes as a large game bullet have well and truely been time tested and approved. Red Deer if you didn't know are much bigger than the average Whitetail in North America but not as big as a Wapiti (Elk). In saying that many Wapiti were also taken by these cullers during the Deer Cull period also. The Lee Enfield as a rifle also has been put through the test of time and endurance and has many strong advocates - if not on this continent then certainly within Australasia.
Although I currently do not use the 303 anymore I personally like this calibre and have absolutely no qualms about recommending it to hunters of all large game.