The 185 TTSX will definitely work, but it is on the light end for a 338 win mag so like others have said you will not be utilizing the full potential of the win mags potential. With that said you will also reduce recoil quite a bit and the high velocity is where Barnes bullets really perform well.
My favorite Barnes bullet for the 338 win mag is the 210 TTSX.
I would use a Barnes 225 gr. "TTSX". I loaded the Barnes .225 "X" for my cousin's .338 Win. Mag. using RL19 moving at 2,900 f/s. .75" 3 shot group at 100mm. These loads have killed 5 oryx in NM. The bullet busted through both shoulders and kept going. All the oryx died where they stood. Good luck.
yeah It will do it but you are only hurting yourself by not using the 210-225.. I shot a bear with the 210 last spring and it worked well. then a couple elk with the 225 and it was a hammer... the 225 seems somewhat light to me even but I think its a good match for the 338 win
" Real elk guns start with the number 3 or bigger and blow two holes, one in and one out." - My Dad