I can't tell you how much I appreciate the help, Valymer. Thank you very much. With your help, I've been able to fix the problem!
The problem was that, after I entered the full string ("oosakakenn") and pressed Spacebar, the IME offered all three kanji together as a [u]group[/u]. Thus, when I pressed what I thought was the right-arrow key (thank you for confirming that the key I was pressing was, indeed, the right-arrow key), nothing happened. Since all three kanji were already highlighted, there was no place for the highlight to go.
I fixed it by typing JUST "oosaka", converting that all by itself and then following up with "kenn" and converting that separately. When I typed "kenn", I got the proper kanji as one of my options. I selected it and, as you said, the IME now remembers it as my preference.
Now, when I type the whole string, "oosakakenn", all the proper kanji appear upon conversion, with the first two highlighted! Fantastic! As to why it had problems before, I don't know, but that doesn't matter now. It's all fixed.
ありがとう!
As an aside, that's a pretty snazzy keyboard! :))