CrossRoads:
Reset needs to be pressed, or held & released, if the USB adapter does not have a DTR, or RTS, output wired out to the Promini's Reset input.
I've tried that pretty much every possible way I can think of and have had no luck at anything with this. I'm still perplexed that it doesn't install, fiddling with reset or not.
I'm beginning to suspect that it really does have no bootloader. In that the cheapest programmer I can find is still at least twice what this cost, I'm inclined to chuck it and start over with some other, less problematic, Arduino clones I've ordered that do specify they have bootloaders and fit up to IDE with no problems. I got this Mini some time ago to build a project I (fortunately, it appears) decided I didn't want to build.
Thanks to all for the attempts at helping, but it looks like this is just a non-starter.