Honestly, I don't think you can go wrong with either of these. You're talking about the best of the best currently being offered.
That being said, you may find one is more "right" for your preferences. The best way to discern this is to go to some meets or a store with a unit you can demo and listen for yourself. This is important especially when talking about Stax, as most of the time these are imported and can't be returned!
Also consider the cost of making sure the rest of your system is up-to-snuff. You can have the finest headphones around, but if your source and amp suck, well, they're not going to sound very nice. Both of these headphones are going to be pretty close to neutral sounding to most ears, but often times your setup will greatly influence how they sound overall.
An electrostatic amp that will do any remote justice to the SR-009 is going to run you, minimum, around $1000 bucks used. A new one will be closer to $2000+. If you're going to go with Stax's drivers, then be aware that reviews are pretty mixed concerning their current lineup. Some feel the SRM-727 is too colored, while others find it fits the SR-009 well. You may be better off looking for a used SRM-717. Or you can try third-party, looking into a KGSS, BHSE, or upcoming Liquid Lightning. These will run you more money (upwards of $3000 to $5000) but are generally considered to be "better" sounding. If you want the absolute most cost effective solution you could perhaps get a used SRM-323A, which is surprisingly competent for what it is.
As for the LCD-3, well, not many folks have them yet, so you may want to wait for more impressions to come (though I have a feeling they're all going to be pretty glowing). You'll also have the benefit of seeing which amp combos people like. One early favorite is the Liquid Fire, which will also run you over $3000 bucks. However there are certainly less costly options like the Lyr (which some people swear by) or the Violectric V200.
So yeah... costly investments, as I'm sure you are aware. My advice would be to take it slowly and don't rush. Definitely listen to one or both if you can, and weight the additional costs of equipment. Pretty standard advice.