If you're not used to shooting heavy recoiling handguns, you probably won't like the 500 S&W Magnum.
Both the 308 Winchester & the 7mm/08 Remington would be great choices for deer. If it was to be used for nothing larger than deer, I would probably go with the 7mm/08. If larger animals were to be targets, I would probably go with the 308. For really big stuff like buffalo and great bears, it would have to be the 500 S&W Magnum or 375 JDJ.
I have a braked Encore 7mm/08 handgun barrel that is quite pleasant to shoot and accurate as well.