I haven't owned any Beyerdynamic headphones so can't comment on how Audeze or Hifiman compare to them, and it's not clear exactly what you want to achieve, other than better "sound quality".
I own Audeze LCD-2, LCD-XC, and LCD-4 and Hifiman HE-560. Audeze have a signature that is a little on the warm side and do bass and mid-range really well, with less emphasis on the highs (which some may find "closed-in" sounding). The LCD-2 are probably the most neutral, the LCD-XC are the best if you want some bass, and the LCD-4 (out of your budget) are the best, period. The HE-560 sound very similar to the LCD-2, but a little more neutral. They do bass well, but are a little more balanced throughout the frequency range.
I would focus more on the type of sound signature you like than the type of driver technology used.
I would add Focal to your list - the Massdrop Elex is really good for the money, very detailed, comfortable and a more balanced sound. I haven't heard them, but the Clear is also in your budget.
If you like things really detailed, see if you can find a used pair of Sennheiser HD800. I think they sound great, but I know they are too "analytical" for some.
Your headphone amp might actually be the weak link. I had the A2 which is the big brother of the A20. It had good sound, but was a little under-powered. There are much better headphone amps available.
I own Audeze LCD-2, LCD-XC, and LCD-4 and Hifiman HE-560. Audeze have a signature that is a little on the warm side and do bass and mid-range really well, with less emphasis on the highs (which some may find "closed-in" sounding). The LCD-2 are probably the most neutral, the LCD-XC are the best if you want some bass, and the LCD-4 (out of your budget) are the best, period. The HE-560 sound very similar to the LCD-2, but a little more neutral. They do bass well, but are a little more balanced throughout the frequency range.
I would focus more on the type of sound signature you like than the type of driver technology used.
I would add Focal to your list - the Massdrop Elex is really good for the money, very detailed, comfortable and a more balanced sound. I haven't heard them, but the Clear is also in your budget.
If you like things really detailed, see if you can find a used pair of Sennheiser HD800. I think they sound great, but I know they are too "analytical" for some.
Your headphone amp might actually be the weak link. I had the A2 which is the big brother of the A20. It had good sound, but was a little under-powered. There are much better headphone amps available.