I really like my T50RP's for this. Nice bottom end bloom and enough detail to pick apart snappy electronic tones, clicks, pops, etc. They could use an amp, but you can get away w/o one, imo. They're well under your limit after modification, though, so it might be enough to buy a headamp in the used market on the cheap, or even an old receiver from Goodwill, a garage sale, what have you.
I have a pair of Denon D950's that sound great with electronic and don't need amping at all. They're sort of a rare headphone because they're discontinued but if you can find someone with a pair to sell you could prolly pick em up for around $100.