I haven't tried an HH AL yet, so that could be the closest answer; but based on my experience I'd say 25th Anniversary or Reflex would do the trick nicely.
It looks odd at first ... but damn is it comfortable. Light and perfectly balanced.
The Morse is also fantastic, it's easily my fave. But there's a whole lot more going on with that guitar, so try both and see if you bond with them.
IMHO, hands down 20th Anniversary Silhouette! My favorite.
That's what I was going to say. Yeah, it doesn't sound exactly like an LP, but to my ears it comes the closest compared to any other MM guitar (including the HH Albert Lee). Definitely may favorite guitar as well.