@stuartk
Apologies if this was mentioned already, but I would consider trying different interconnects between the Aqua and the H390. In your A/B tests, that is an additional change you are hearing (with and without the IC's).
Like your Aqua, I too have a R2R dac that uses vintage ladder chips known for natural sound with delicate and not bright highs. I had sibilance issues of different degrees with a number of interconnects that were well regarded (Audience, Cerious, Audio Envy, Black Cat, others).
Eventually I found the right synergy with Zavfino Fusions IC's (silver and OCC copper).
Adding power conditioning to my system, (puritan psm) and different power cables (triode wire labs digital american) was another improvement in the high frequencies.
As to your original question on what dac designs to consider if you go that direction: I'd stick with R2R, and look for designs that have intentional focus on a quality output stage and power implementation.
How to proceed?
Removing my Aqua La Voce dac from my system has eliminated fatigue and reduced but not fully eliminated sibilance. Running my Jay’s CDT2MKIII into my Hegel H390’s onboard dac is definitely a more relaxed presentation but tonally leaner than I prefer. Some have suggested I swap out the Hegel, but first I’d like to try another dac.
What I’m unclear about is how to go about choosing another dac that will not duplicate the same drawbacks I’ve experienced with the Aqua. Are there details in the design or specs that can guide me in this regard? I'm unsure how to proceed.