.270 isn't a short action caliber.

Hard to go wrong with either the 7-08 or 308 at those ranges. I'd run heavy bullets for either & go with the one that shoots best (or maybe the lighter I did there's hiking involved)

Cant go wrong with 7-08 or 308 if you are willing to handload.  Both have decent factory ammo avail that would perform well.  The Russians have been shooting red deer with worse for 50 years. (not an excuse for underperformance calibers though).

I was just reading the 34th anniversary edition of gun digest (1980) and and John T. Amber was describing his hunt for red deer in Russia in 1978 or 1979 and they gave him an auto-loader in caliber similar to 9.3X53R to use.  I gather that he was not impressed, but both rounds you contemplate are soooo much more capable. 
pm me if you want a scan of the article.

id say the .270 would do the job no problem, ive used them for Red stag an big Sambar stags, nice solid projectile will work fine.

failing that the 308 never fails to get the job done!