Budget or Mid-Market DAC for Use with Naim HDX

Has anyone used a mid-market DAC with a Naim HDX? I’m using an HDX with an Audio Alchemy DAC in a Box. The sound is almost as good as a Naim CDX2, but not quite…

My question is whether a more recent mid-market DAC that is >=$1000 will enable the HDX to surpass the sound quality of the CDX2. I checked the Naim Forum, and all HDX/DAC-related traffic was about the HDX and nDAC combination, which is way beyond what I’m able to budget.

Does it make sense to find a DAC that will get me close to the sound I had with the CDX2? Alternatively, does it make more sense to return the HDX, and retrieve my 2004-era CDX2?

My system is: Linn Exotik Pre and Akurate 2200 v2010, HDX, and AA DAC. ICS are AQ Cheetahs. Digital coax is Transparent Premium.
The Naim crew were pretty big on the Lavry DA10 so that would be worth looking into.