You may be able to find a used Sony TA-ZH1ES within your budget. It's a fantastic headphone amp with a really good DAC. I have about a dozen headphones, many of them "high-end", but one of my favorite pairings is with my Sennheiser HD-6XX. It doesn't support MQA, but the sound quality will be much better than what you have now and you won't miss MQA.
Of the ones you've mentioned, I owned the Mytek Liberty (not the II) and found it very analytical and digital sounding. Its headphone amp is not close to the Sony. The Bifrost 2 is a great value for its price, but no headphone amp and it also has more of a digital edge to it than the Sony.
Of the ones you've mentioned, I owned the Mytek Liberty (not the II) and found it very analytical and digital sounding. Its headphone amp is not close to the Sony. The Bifrost 2 is a great value for its price, but no headphone amp and it also has more of a digital edge to it than the Sony.