A good tube hybrid or solid state is your safest bet. Melos SHA-1 used falls into that category, a great amp for low-Z cans. As far as solid state circuits go, I generally prefer circuits with 3 OP-amp channels (L/R/GND) and dedicated current buffer. So circuits like the PPA, M^3, or even a Pimeta run at higher voltages are your best bet... for starters. Stay away from the over-priced single OP-amp designs like the RA-1 (for example).
As far as tube circuits go... OTL amps are hit or miss, more misses than hits. The K701 needs current and voltage reserves to effective drive them, so that effectively rules out most OTL designs. Transformer coupled amps are another ball game, where these cans are at their sweetest. The problem there is that none exist under $500, unless you are willing to pursue a DIY output transformer solution. That was my K701 fix. Got a DV337 used for a very good price with some great sounding tung-sol tubes, and DIY'd an output transformer coupler for the amp. The entire package was around $600.