Is it possible to buy a decent tube amplifier in the $2000-3000 range?

I would not consider a speaker with a 86 dB sensitivity and a load of 6 ohm nominal, 4 ohm minimum as a "very efficient" speaker.

I would think that you would need a powerful tube amp (100-200 wpc) to drive these speakers well, with tubes.
I am not aware of any powerful tube amps costing less than $3K, sorry.

I don't believe that a 40 wpc tube amp will do those speakers justice.
They probably work best with your NuForce amps.

I’m beginning to think I need about 200W per channel, as my 160W monoblocks are not quite doing the job.
Maybe the way to go is solid state at the price range I’m looking at ($2000-3000).
Any opinions on which way to go?

Well, if 160 watts is not quite getting the job done, 200 watts won't give you much more, maybe 1 dB if you're lucky.
I would think you would need 300-400 watts to drive those speakers.

300-400 watts, for $2-3K, yeah, you're only choice is solid state.

Or, as Douglas and Shardone have implied, get different speakers which are easier to drive.