Just going to give a few random ideas, nothing concrete. I honestly don't know how many people are familiar with psybient so we'll see how many responses there are.
I used to have a pair of ATH-AD2000's - those were quite strong with electronic - fun although colored midrange, deep but not massively thumpy bass, extended highs, fast perceived speed (partially due to the coloration). Soundstage was good but not great. You should be able to pick up a pair for around $500 or so, maybe a bit less. As far as mid-fi phones go (looks like that's what you're aiming for) I also owned a pair of HD-650's which I found too dark for psybient and also unimpressive in their imaging and soundstage size. I'm not experienced enough with Beyer's mid-fi phones to comment, although I would stay away from the 770's (awful v-curve frequency response - used to own a pair) and might consider the 880's. I haven't tried the 701's specifically with psybient but would suspect they would be too lacking in bass.
If we go quite a bit higher in price, you can do better. HD-800's are excellent for psybient, in my opinion, as long as you don't find the bass anemic (I don't, and I've listened to them on multiple occasions, multiple rigs). Personally, the best I've ever heard Shpongle was with a pair of Omega II mk. 1's, actually out of a Stax T1 amplifier, nothing fancy. The layering and imaging was absolutely perfect, no details lost, and the speed was faster than my JH13's which I consider to be quite fast. The flute and other physical instruments sounded extremely natural. Of course, a pair would be out of your price range and I would strongly advise you hear a pair first, since they are on the dark side of neutral, but that was my experience.
I won't give any amplifier suggestions, although personally I would go for a nice transparent SS design - an M3, CKKIII, Gilmore lite, and the like will power any non-k1000 phone to transparency, although they won't pleasantly color it. The new Schiit design looks interesting but I will refrain from commenting since I haven't heard it.
By the way, in any case, try to find a meet in your area and go listen - that's the best way to prevent yourself from being disappointed.