$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.
