I have an ASAT. I bought it several years ago from a shop that had a bunch of used G&L's, so I played them all, without knowing much about them. I kept coming back to the ASAT, and it came home with me. I just loved the neck. Best neck for my hand that I have ever had my hands on.
Of course, I was sitting down in the shop, so I did not notice neck dive. To be honest, I have a FrankenJazz (a modified SX) and a Fender '74 Pbass. Both of them had neck dive that was at least as extreme as what the ASAT had. Strap pretty much cured that. On the other hand, the fact that the strap attached closer to the bridge meant that I had to reach farther to play low notes. I am sure that it's the same with Firebirds or Les Pauls, and was not that big a deal but I searched for a solution...
Long story short, there used to be a guy called "ASATman" aka Dan Pierce who sold a strap extender just for ASATs. He milled these out of aluminum. I bought one from him, and it turned my ASAT into the best balanced bass I have ever had. Not sure what's become of the guy. Rumor has it that he sold off his ASATs (he had a huge number of them), and no longer visits the forums. Not sure what the story is on that. If you get an ASAT, and are bothered by neck dive or playing position, I would be happy to give you the contact info I have for him, but it's been several years and I don't know if it's still good.
I have messed with my ASAT enough to love it, and they are definitely cool looking basses. If you get one you'll get lots of positive comments about that. I think they tend to be a pound or so lighter than the L2Ks because of being a smaller body, but G&L has used so many different woods that YMMV mucho.
To me, the L2Ks have the edge on comfort due to playing position and the contoured, more rounded body. I like the stock necks on the ASAT, and that's where you spend most of your time... So I probably would not get an L2K unless I could get it with a #8 neck... But there are lots of people who like the wider neck on the stock L2Ks...you might be one of them.
My advice is to play both, and take a strap to the shop with you when you do
