I don't think some of those places have "other sectors" that can be shifted to. I'm not arguing for burning fossil fuels, but shrugging and telling people already past the edge of desperation to "adapt" is cruel and unreasonable.
The highest dependency on mineral resources is 16.1 % and it seems unlikely that the entire sector would get killed, i.e. if the asteroid contains mostly iron and nickel, then only those parts would be affected. Without running the actual numbers it's hard to tell if some countries would get hit unreasonably hard. But even if, we could still opt to support them. We could have a tax on space mining operations and spend the proceeds on support for affected countries. So this should at best be a question of how, not if.
By the way, I don't think space mining will be of any practical relevance in the foreseeable future. I would also much prefer if we would learn to operate economically with the resources we have instead of just reaching for more resources. So I have no real skin in the argument, I just found this position against space mining unjustified.
I understand where you're coming from, I'm just enjoying the discussion as well. I think you're right, it's definitely feasible that we could ease the impact of such mining, but I worry that we wouldn't. The countries you've listed are ones that most people shrug and feel badly for, but don't really do much to change. They've been basket cases for longer than most have been alive, and the reasons why are complicated and largely older than anyone alive.
I worry that it would be cheaper and more profitable for some to simply write them off. You're right however, that it's not a problem with space-mining, it's a problem with people.