Well there's no gap or overlap between the tiles, so I'm not sure that's the issue. I'll keep comparing my code to vanilla code and see if I snafu'd something somewhere.
In terms of progress, however, I've arrived at the point that when opened, a cleanly formatted GUI is displayed that lists every relevant piece of equipment in the player's inventory (including equipped items) as a button. Basically, you'll click on the gear you want to enhance, go to a second, item-specific GUI and expend some experience to purchase abilities or perks that you've unlocked. It should be really fun when I'm done.
That issue aside, I haven't been able to trigger any of this with a keybinding, because there's no way that I know of to get a player reference that isn't null, and therefor I can't access its inventory, either... Quite a few mods have done it, though, so there's got to be a way.