Former military chiming in here. Yes I would take the shot, .308 (7.62x51mm) 175 grain round. It would do the job and them some. I would not go past 500yds with this rig partly because I don't have many chances to shoot long range anymore. If I did and practiced at those ranges I'd take the shot. Either with the .308 or something in the 300WSM ranges to include 30.378 cal. Problem with these is if you do a lot of shooting about 2000 rounds the barrel is shot out. The .338LM is like that. I'd like to try the 6.5 Grendel stays supersonic at 1200 meters and less affected by the wind also. That 6.5 would cause me to go back to the long range meets I use to go to.

Anyway .270 cal would be good, 30.06; 6.5x55 Swedish would also fit the bill.

Now someone has mentioned ethics and hunting. Most hunters don't need to shoot past 150yesbecause they do not take the time and practice, practice, practice. Shot placement and the willingness to shoot in all different extremes and weather goes a long way. The two biggest obstacles are accurate range estimation and reading the wind. If you can't do that don't go past 200 yds.
The only way the shot becomes unethical is if the rifleman/hunter does not do his job and does not pass on the shot. Think you can’t hit at 300 yds and can’t do it on paper? Move closer to the target till you can hit the bulls eye consistly. That’s as far as you should shoot. Every one is different, one hunter should not shoot past 200 and for another it may be farther.
I routinely take deer at 150 to 200 yds with my .223 69 to 70 grain rounds and never had to track one down. One shot, one kill. Same with the 30-30 and both my .308's.

In the end stick with what you are comfortable with.