Sub-$1250 DAC w/ built-in Headphone Amp

i have my fx tube dac still... after reading the stuff about the shootout in arizona -- bought it stock sounded very musical but somewhat rolled off and veiled with good tubes in it (stock chinese tubes turn up the treble some, but a little grainy) then sent to chris j/wally to upgrade level 2 mod - comes back cleaner more transparent more extended - sounds right with proper, good old stock ax7’s in it but the treble can still be a lil problematic (sibilant) now that caps and wiring have delivered excellent extension - maybe the thing needs a thousand hours of break in lol...

sibilance not as bad as dena ares2 or bifrost mb, but still there on some material - tried various cabling...

so overall i like the mhdt and audio mirror (with tube upgrades) more, or the lovely older mf trivista or ava fet topp dacs more than the fx for ease musicality and insight without force feeding treble detail -- but i haven’t thrown in the towel yet on the fx

i am not a headphone guy so can’t comment on how good the hp output is... but that lil volume control is not the greatest... imo best to leave it on full blast for most clarity and control volume elsewhere

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.

It's a lot cheaper than what you listed, but I would maybe consider the Topping DX7 Pro.  It uses very good clocks (the Accusilicon).  Then replace the two headphone amp op amps with Burson V6 Vivid and you have your discrete output stage.

How about a used Ayre Codex?

@oldears - not much love for the Codex here: https://www.audiosciencereview.com/forum/index.php?threads/review-and-measurements-of-ayre-codex-dac...