I think there's a higher likelihood of getting a keeper if you're going EII/ESP versus going LTD, especially if you're ordering online. If you've got a shop local to you that carries a bunch of LTD's just go in and play them until you find the best of the bunch and go home with it. If you're buying an LTD online new make sure you can return it without a charge because the last couple I've picked up I've been underwhelmed by (one was dead sounding and had zero sustain, another had some extremely sketchy finish work to the point where I wasn't sure how it made it out to a store with how crappy it was). In my experience (which isn't vast) most of the newer LTD's at GC kinda feel like toys (super light weight, zero sustain, corners cut on the finishing, hardware can be sketchy) unless you're getting one of the ones that are $1300+. With the LTD new pricing now a days you can get an EII or ESP used cheaper with a case a lot of the time.
I'd go for a used EII Eclipse and try to get something sub $1200 which I think is still doable, might be able to swing closer to $1k if its off a forum and you're willing to deal with some wear and tear. Gone are the days of like $700 ESP/EII ebay/reverb finds unfortunately.