Sequis Motherload - speaker emulator and dummy load - is exactly the thing for the job. The problem with a lot of speaker emulators is recording high gain sounds as well as clean, and the controls on the Motherload make both easy to do. This article covers it pretty well.
http://www.soundonsound.com/sos/jul05/articles/sequis.htm
Its not cheap though, and you do need to be prepared to do the tweaking to get an exact sound like the LSS. Took me about 3 weeks of fiddling to get the overdrive to exactly the same tone. The clean I got as soon as I got the machine.
Also works for any other silent recording, like bass amps.