Tube preamp match with Solid Stare amp

Also a Backert Labs fan...currently have a Rhumba Extreme 1.3 and sounds beautiful with either my Pass Labs or Levinson amps.

Audio Research Ref5SE should be a decent pairing. But 10k input impedance is minimum and borderline. I think you may actually be better off upgrading to a very good solid state preamp in your case. 
 

I am using the Rogue Audio RP-7 balanced tube preamp with excellent results; I previously had an RP-1.  I was using it with a Benchmark AHB2 SS power amp which gave excellent results, but switched to a tube power amp (Rogue Stereo 100 Dark), which I believe resulted in a modest "downgrade" in the overall sound quality .... perhaps too much of that good tube stuff.  So, I am going to switch back to SS power amplification, most likely with Odyssey Stratos Extreme ("plus some added upgrades) mono bloc amplifiers.  The RP-7 is excellent, with the low impedance you seek, and at roughly $5K new it fits in your budget. I have had several questions that I directed at the folks at Rogue Audio and received prompt and helpful responses.

FWIW, I run a McIntosh C220 (discontinued) into an Arion Audio S500 switching amp. Love this combo. The C220, which you can find used without too much difficulty, has the features you want and more. It’s very good tube phono stage is MM only. Moving up to a used C2300 or C2500 will get you a MC/MM phono stage, but might bust your budget. I think my C220 has balanced ins and outs for everything except the phono stage, which is unbalanced. Also not sure if the C220 is internally balanced or not. Also, I have about a 14 foot run of interconnect to the amp with no issues. My amp is neutral and very clean. The C220 is just a bit on the warm side of neutral.

Buy a BAT. They’re truly balanced design’s, and they have many different preamps available in your price range. Research Balanced Audio Technologies. They’ve been around since the early 90’s and have an excellent reputation with fantastic customer service if you ever have a problem. 
 

I’ve used BAT preamp’s with nary a problem and each was a joy to listen to. For up to $6k you can acquire a VK51 with the Russian Super Tubes and still save some money.