I would say an ALHH with the solid Rosewoood neck is the closest to a Les Paul in the EBMM line. It has the same pickups as the Axis but the wood choices make it sound different, at least to my ear.
+1 for the Al HH. Additionally more easy to swap PUs on that one, which will you get closer to a LP. But it's not unlikely that you don't want to swap PUs after trying the Al HH.