My son has been shooting the 85 grn Barnes TSX in his 243 for about the past 6 seasons. Mule deer and antelope 100 yds and out to 250 yds max distance so far.
I realize the difference between the 6 BR and .243 Win is quite a bit in powder capacity, but I would assume you could still get approx 3000 fps with the 85 grain bullets?? Should be enough for the bullet to do it's job. The Tipped TSX may be even better at the lower velocity of the BR.
The Barnes have worked great for my son, and they've proven accurate. His rifle is just a Browing lever action with a cheap 4X scope, heavy trigger and very thick crosshairs..........still shoots 1 minute at 200yds off the bench.
Main reason I mention the Barnes, is the 20 yds distance you mentioned. Alot of bullets may not hold up and penetrate shoulder bones at that close of a distance, especially the thin jacketed type. IMO, a bullet blow up is likely at such close distances if any solid bone is hit..........the Barnes will not blow up.