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

$2K to $3K is plenty for a decent tube integrated. Raven, Rogue, Line Magnetic, Melody, Prima Luna and Cayin all make competent products.

IMO, Cayin amps are the real sleepers. Performance wise, these amps are basically on par, if not better than the LMs and Prima Lunas. Chassis construction is often beefier and they have neater point-to-point wiring than the others. The downside is they have almost no dealer network in the U.S. at the moment. If buying used, and warranty isn't a factor, consider a used Cayin, which could be well under budget.

Rogue tube amps use PCB mounted tube sockets. This factor alone was enough for me to avoid the Cronus Magnum. Installing and removing tubes in these amps makes the PCB sound like a bowl of Rice Krispies. It makes the whole board flex in quite a stressing way. They also have audible transformer hum IME.

The Rogue Hybrids don't achieve any of the tube magic, despite what their marketing might lead one to believe. I owned the Sphinx and Pharaoh and neither sound anything like a tube amp. The Cronus is far better than the hybrids.

I went hybrid  Rogue RP-1 Pre (tube) and PS Audio S300 class D amp.  Wyred4Sound also has a dual mono Class D that will get you in at about $3K with an RP-1 

I own the Rogue Cronus Magnum II with KT-120 Tubes. For $2,495.00 you can't beat this amp. Sounds much more like a $5,000.00 to $8,000.00 amp and the KT-120's will drive many speaker varieties
 with ease.

I'll throw in another vote for PL and Rogue. I've owned and auditioned several pieces from both - they both sound great in their own right, IME PL more fleshed out in the midrange, Rogue more dynamic, wider bandwidth, better bass control. Within the Rogue line, the all-tube vs. hybrid are split similarly, with the all-tube designs having a warmer, more complex midrange, while class-D hybrids have iron-fist bass grip and bass control.

That said, an option at the top your budget would be a used pair of Rogue M-180s. I've been running these in a system for a few years and love them for their ability to naturally and effortlessly handle many styles, with relatively inefficient speakers (ProAc 2.5s) They're also fairly tunable via tube rolling, especially via the 12ax7 phase inverter.  

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?