Sweet. After rereading OP and your post, I believe I misinterpreted it. But I will say I have that problem too...
Problem A: What OP and you are talking about: When you switch computers to one that doesn't have the bookmarks, you have to click through every one to get the icon to populate.
Problem B: Once FF has an icon, it will never update it. Even if you delete and re-add the the bookmark. I have no workaround for this.
It just opens every bookmark to load the favicon. Brute force, but it works!