Tube Amps With Good Bass Qualities

The M-125 has excellent mid-bass. I roll mine off ≈88Hz, so can't comment on the low bass. Square wave is very good, so I'd imagine the low bass is fine.

As built by VTA, the M-125's were very noisy, but can be quietened quite a bit. In my case, almost 40db.

Complete instructions:
http://ielogical.com/Audio/VTA_M-125.php

Line Magnetic, absolutely insane performance for the money. LM 518ia is long time Stereophile class A integrated and the LM 508ia has even more oomph and more dynamic.

Agreed on the Line Magnetic. The LM-805ia - which has replaced the 508ia but with similar specs - lists for the OP's 5k price point and has a negative feedback control with 4 positions.

Manley Mahi Mahi mono blocks.  These are scary good providing a solid foundation for bass.  They are very quick with EL 84 tubes and get the midrange right.  These blast away so many stereotypes falsely attributed to tube amplification.

Another vote for Atman-Sphere.  I have the M-60s with the upgraded power supply driving speakers that are basically 10 Ohms (minimum is about 8, I think), and the bass is deep and articulate.   I couldn't be happier.  But as has been well articulated above, you have to have a happy match between speaker and amp.   If you don't get this right you are going to have issues.  BTW, these amps are gems from top to bottom.  Get the right tube complement and these amps sing like angels.