First question I have when you say SBR, exactly how short are you talking?
In the "pistol" length barrels, somewhere in the 10" range, 300 BO is going to have nearly double the kinetic energy with a 110 grain bullet vs a 5.56 with a 55 grain bullet.
IMHO, between the two, 300 BO all the way for an SBR. I know, I know, some have issues getting their 300 BO shorty to run right, but I've seen short 5.56's with issues too. If you don't reload at all, then 300 is going to have the issue of finding ammo and when you do, it's more expensive. If you do reload, then it's a different story. 5.56/223 cases (which the 300 is based on) are stupid easy to convert to 300 (and there are frequently tons of 5.56/223 brass laying around on ranges). I've got a pretty decent stock pile of 300 ammo and to date I haven't spent a single dime on 300 brass or commercial ammo, just powder, bullets, and primers.
Physics is physics, there's nothing magical about 223/5.56. It's a 22, it sits somewhere between the 222 Rem and the 22-250 in terms of velocity. It has a narrower range of bullet weights/types than 300 too. If you are going for HD, then 300 is the choice. You get more energy on target and a much broader range of bullet weights and types. I'll take a bigger, heavier, expanding/fragmenting type bullet for that scenario any day.