I'm not sure if it's reasonable to expect to "replicate" what you have in a headphone system, the experience of listening to music through speakers and headphones is very different. Your budget allows for some decent components.
If you're willing to buy used, consider the Audeze LCD-2. They are detailed, but on the warm side and have good bass and will sound great with the types of music you enjoy. You can get some new earpads for a set of used of headphones if you're squeamish about buying used. The Audeze ones are about $100, there are some nice aftermarket ones out there for less.
If you prefer new, check out the Hifiman HE-560. They are similar to the LCD-2, but are harder to drive and will require a more powerful headphone amp. Another great set of headphones is the Focal Elear (or Elex from Massdrop). These are more "neutral" than the other two I've mentioned, but are very detailed and easy to listen to.
An outstanding headphone value that I recommend looking at is the Massdrop Sennheiser HD-6XX headphones. These are super comfortable and sound great and are only $200.
Like speakers, the headphone is going to impart more to the sound signature than the amp, but proper amplification is important. I don't have any suggestions in your budget range, most of my amps cost more than that. If you can find a used Musical Fidelity M1-HPA or M1-HPAP, that's an inexpensive amp that I have owned that has a lot of power and sounds good.
If you're willing to buy used, consider the Audeze LCD-2. They are detailed, but on the warm side and have good bass and will sound great with the types of music you enjoy. You can get some new earpads for a set of used of headphones if you're squeamish about buying used. The Audeze ones are about $100, there are some nice aftermarket ones out there for less.
If you prefer new, check out the Hifiman HE-560. They are similar to the LCD-2, but are harder to drive and will require a more powerful headphone amp. Another great set of headphones is the Focal Elear (or Elex from Massdrop). These are more "neutral" than the other two I've mentioned, but are very detailed and easy to listen to.
An outstanding headphone value that I recommend looking at is the Massdrop Sennheiser HD-6XX headphones. These are super comfortable and sound great and are only $200.
Like speakers, the headphone is going to impart more to the sound signature than the amp, but proper amplification is important. I don't have any suggestions in your budget range, most of my amps cost more than that. If you can find a used Musical Fidelity M1-HPA or M1-HPAP, that's an inexpensive amp that I have owned that has a lot of power and sounds good.