UPDATE: Received the new set of pads from StopTech to test with the new calipers -- same problem. In most fitment cases the pad backing plate was just slightly too wide to fit between the caliper abutment plates. With a certain pad in a certain sides of one of the calipers, I was able to get the pad to fit, but even with the caliper resting on the floor and pressing down on the pad, it took quite a bit of effort to get the pad all the way into the caliper. If the caliper had been upright and mounted on the car, getting that done would certainly have required a mallet, but even after I got the pad in, it took a mallet to get the pads back OUT, which suggests to me that even in that one case that fit into the caliper, the pad would have bound inside the caliper if I tried to use them on the car without modification. Meanwhile, the race pads that my shop had ground down to fit in my existing calipers slid into the new calipers perfectly.
At this point I'm going to return the replacement calipers rather than going through the cost and hassle of swapping since it won't fix the issue. As to the issue itself, I guess at this point I have no choice but to hope that this problem will go away with time and use, and that pads will start fitting properly without needing to be ground down first. If that happens, great. If not, there isn't really a different BBK on the market I'd rather have, and I'm not crazy about going back to stock, so I guess in that case I'd just keep what I've got and consider pad grinding an unexpected but unavoidable inconvenience of the kit.