The .270 Winchester is arguably more powerful than a .308 win. So go with the .300WSM if you want to see any real difference.
The .270 pushes a higher sectional density bullet faster compared to .308 win
.277" (.270) 150 grain, SD .279
.308" (7.62mm) 180 grain, SD .271
The .270, with a stout hand load, you could push a Nosler Accubond 'Long range' bullet fast enough to end up with roughly 1735ft-lbs energy, 2282fps @ 500 yards. Which should be more than adequate if you're a good enough shot to ethically hunt at those ranges. SD is high enough to penetrate CXP3 game, and the Accubond tends to have the best expansion right around 2300fps.
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