Best Tube Integrated

I would keep my eye out for a Jadis Orchestra Reference, JOR, on Audigon which should come in under your "less than" $2K price point. I puts out about 50 WPC and particularly with KT 90 tubes has a lot of punch but still a lot of tube sweetness.

Aronov LS-960I, now available only used.
Provides your requested "big output with a lot of bass," as well as "having some juice."
Specifically renowned for enabling forceful, faithful-to-the-music bass, limited only by a speaker's abilities.
Sounds very natural, easy to listen to, realistic, with music free of any dryness, or listener fatigue, rather than specifically warming up, or sweetening.
Does allow recordings to sound their best, and definitely avoids what is less captivating about many solid state designs.
One of the very few modestly priced tube designs that delivers the kind of profoundly satisfying bass that can thrill the audiophile heart.
From the way you described your listening preference, any choice that fails to deliver great bass might be a short-lived experiment.

DK Designs VS.1 Reference Mk.III integrated amplifier.
Heard this peace on a few occasions. I was really floored when I find out it was around $3000 new. However take my recommendation lightly because I never had this peace in my system for evaluation. But if I was looking for a integrated amp it would be this or Manley's Stingray.
In your system, DK would probably be a better choice.
Cheers