It seems you have a pretty nice setup now; one thing that can negatively affect low volume playback is if your analog volume potentiometer (I think the AVA has an Alps Blue in it) is very low to the left - meaning below 9 or 8am on the dial. That would mean the audio is running through a lot of the carbon resistance track on the potentiometer and potentially losing dynamics on the way. The newer AVAs have a "low gain" button, that should give you more range on the volume control and better sound at lower volumes.
You could test this with your DAC, if you can control its output. Set its output to very low so you have to turn your volume on the AVA pre way up and see if you get better sound.
If you get an integrated, I would think getting one with a digital style volume control (with a resistor relay ladder like on the RAGNAROK or equivalent) would do better at high attenuation than the Alps Blue. My son had the same problem as you, with a lower budget than yours and he has high sensitivity (94dB) speakers with an older amp and an analog volume pot that he could not turn up past 8am on the dial and his sound suffered. We tried an Arcam SA-10 and it was night and day. He was amazed at the difference in low level listening from the Arcam. The Arcam's, the Creek Evos, Audiolab, etc have the volume control in a chip - something you might want to look into.