Clean and powerful SS amps are typically good matches for the LCD2.
I can only recommend what I have listened to for a long time. I suggest two relatively affordable amps: the Vioelectric V200 and the Schiit Mjolnir, both under $1,000 new and, I hope, in your price range. You can pick up a used one, in excellent condition for a lot less than $1,000, especially now that many are upgrading to the bigger brothers (V281 and Ragnarok).
Of the two, the V200 had a cleaner sound and power enough for the LCD2 but you could get even more power with slightly less sophisticated sound with the balanced V181: thunderous bass but somewhat dark sound. To stick with the Schiit family, the Mjolnir can make the LCD2 really shine: good bass and clear mid-range with much of the dark sound of the LCD2 gone. But, depending on your sensitivity, the sizzling treble may prove to be fatiguing in long run so it's best to audition first.