BAR in .338 win mag should be a sufficient cannon!
I own a Bar in 338, its a brute for recoil and i shoot Rum calibers. I have tried recoil reduction by least money....a good pad and a mercury tube. So now i will remove the sights and spend the money to put a break on it.
I have a 30-06 Bar and it does not even have a pad, just the recycled hockey puck that come on them. Yet the recoil is fine. By happy coincidence having inherited another 30-06, its for sale in the EE
Your options are Remington 740 family, Browning Bar, and Benelli R1...in increasing cost order. The Benelli is pretty expensive and not out long enough to find a significant drop, used. I have owned three Bars, they are good, accurate, simple guns. i have never owned a Rem auto loader, some have a reputation for poor accuracy.
I see from one of your posts that a 300 bit after 25-30 rounds. If you are wanting a target rifle or range toy, semi hunting rifles are not really suited. That is where a M305 would come in, shooting surplus cheap...but they are a legitimate 2 MOA rifle. Any semi set up for hunting is going to have a sporter profile barrel, and groups will open up quickly if you shoot rapid fire strings.