Integrated Tube Amp approx. $2,500

Look up Lejonklou Boazu. It’s an amazing sounding integrated and the used price is exactly in your price range. It’s dead quiet so perfect for high efficiency speakers like your Cornwall, detailed and not fatiguing at the same time. It’s very minimalistic in design and features but it still has remote control. 

+1 on the Octave 40SE. 

If you only need an integrated for component switching and not a DAC or phono stage, I would go with a small builder like James Burgess.  Ask him to install a selector switch, shouldn't be  hard.  I just had a Burgess and it was a very very nice sounding amp.  Lot's of small builders, including Had, could probably get it done in your budget.

I’ve owned Line Magnetic (good, especially for the price), Raven Audio (quite a bit better SQ than LM) and Rogers High Fidelity (run away, run very far away).

I have a Black Ice tube DAC that I like a lot and is a bargain, so I can speak to their quality, value proposition and customer service (all great) but I have not heard their tube amps. 

In my systems, to my ears, the MOSFET-based Yamaha integrateds (especially the 2100 and 2200) outperformed all of the tube and hybrid integrateds I’ve owned, that includes the popular Rogue CMII and Pharaoh amps.

To my ears it has more of the tube strengths than most tube products I’ve heard/owned, without the typical drawbacks. 

The only tube amp I personally know of that gives them a run for the money (on balance) is the Cayin A88T. 
 

If you’re set on tubes, you might as well look for an integrated employing 300B tubes, since your speakers are quite efficient and don't do bottom octave bass anyway.