Try some 110 Accubonds ahead of RL 22. Fed GM215M primers.
Just read an article by Terry Weiland (who does not condone such) relating a much earlier magazine article by Roy Weatherby hyping the virtues of the 257. On an African trip in the 50's Roy took a considerable assortment of larger plains game, including a Cape Buffalo !
While most PH's (and some of the licensing countries in fact) wouldn't allow the 257 on any dangerous game, ( most have a .375 minimum) it is certainly quite capable. Many have used the 257 Weatherby satisfactrily for pronghorn, deer, moose and elk.
It will certainly be most adequate for Caribou out to 250 yards,
but then again, the 250 Savage & 257 Roberts 100-grainers would most likely be as well !) A well placed shot with a properly constructed premium bullet is key.