Alright - I'm going to respond before reading the other responses, so if I duplicate what others have said, sorry. If I say something that contradicts what others said, it just goes to show the complexity and subjectiveness of the malt.
First off, you have great taste and a good eye (tongue?) for whisky. As I was reading your post, I would think of various malts and lo and behold, you would mention them!
I was thinking Uigeadail would introduce you to a sherried Islay, but alas, you have already tried it.
I was thinking if you liked Laphroaig, you owe it to yourself to try Lagavulin 16, as it is divine. Alas, you already mentioned it. This is a no-brainer. Get it. If you REALLY liked Uigeadail, try the Lagavulin Distiller's Edition, as it is also sherried. However, it's pretty expensive...
Definitely try the Laphroaig Cask Strength. It is an obvious choice, as you clearly prefer the strong phenols of Laphroaig, and this gives you what you like in a whole new expression. I started off with Laph CS, and never looked back! Good stuff... FYI- don't be afraid to pour a CS whisky over ice, as it can handle it. Weaker whiskies fall apart over ice, but the big CS whiskies stand up just fine, and as the ice melts, it gives you all new flavor profiles.
For something different, find a bottle of Aberlour A'Bunadh. It is NOT an Islay, but it is a FULL POWER whisky! NOT for the faint of heart. It is a ~60% (batches vary) cask strength, 100% matured in sherry casks. It is sublime!
HP 18yr is one of my favorites, but the 12yr is not even close. Ok, it's close, but it's just not there. The 25yr is fantastic, one of the bests ever, but it is WAY too expensive for your purposes here. Dalmore 15yr would be a better choice than HP 12yr if you are looking for something different.
Bunnahabhain is a "light" Islay. It packs a bigger punch than HP ever could, but nothing like Laphroaig. I have enjoyed the 12yr expression, and a friend loves the Bunna 18yr (which is pretty inexpensive for an 18yr).
A favorite of mine, if completely different from what you are focusing on, is Glenfarclas 15yr. Not sure if you can get it in Japan, but it is one of my favorites.