Audeze vs. HIFIMan vs. ??

Thank you for mentioning earpads.  I've bought many used headphones and that's always the first thing I do is replace the earpads.  I've also done that with most of the headphones I've bought new and it has resulted in better comfort in every case.  A lot of earpads are thin and your ear may be on or very close to the driver, which can affect the sound and comfort.  I haven't noticed a huge difference in sound as a result of changing earpads (it's hard to A/B!), but I know many people do.  It's worth it for the comfort alone.  Audeze has some of the best stock earpads in my experience.

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.