If it has to be an integrated, a hybrid tube/SS model like the Pathos Logos or Unison Unico 150 would likely sound beautiful with your speakers. Upscale Audio has a used Unico 150 available now for $3300 that would obviously save you quite a bit of $$$. But another possible outstanding option is a Vitus RI-100 integrated on US Audio Mart for $5750 that retailed for $13,000.
I’d opt for a separate DAC rather than relying on one built into an integrated. To me, the DAC is far too important to leave to chance or be tied down to one built into another component. Given your music and sonic preferences, I’d strongly consider a tubed R2R NOS DAC from the likes of MHDT or AudioMirror.
For a streamer I’d get an Innuos Zen Mk3 that can also rip and store your discs so you can have access to all your music from your chair. Hope this helps, and best of luck in your search.