Best DAC+Volume Control under $1000 (used)

My vote is for the RME ADI-2 Fs.

Chord Qutest if using the receiver as the preamp since you only have single-ended inputs.
I've only heard the Qutest and it's a nice DAC. I've got a Benchmark DAC3 which is pretty close to the RME. I've used it direct to the AHB2 as well as through the HPA4. Sounds great.

IMO, I'd opt for transparency at the source and fiddle around with tone using preamp / amp.

what about using a straight DAC without volume control, and just connecting it to my Receiver CD-input, then pre-out to my Amp?

Oh good God NO!  Running the Zen and a decent DAC through an AVR’s preamp section is heresy and will destroy their goodness.  Get the best DAC with volume control you can now and add a stereo preamp down the road if needed/desired.  The Soekris 2541 or Gustard X26 are both great options if you can stretch a little bit. 

If you're considering Denafrips vs. Chord, I believe Qutest price is closer to the Pontus II and about double the Ares price. The Ares is the only one of the three you're likely to find under $1K. Cheers,

Spencer

Ideally, you would use a hi-bit (such as a 32 bit) DAC with built in digital volume control, with an output voltage that is similar to the input voltage of your power amps sensitivity for full output. The digital volume controls on these newer high bit DACs are much better than the digital volume controls on the older lower bit DACs that often led to bit stripping which compromised sound quality. The standard voltage output for CD's is 2 Volts, for HDCDs 2.2 Volts, single ended via RCA's. While true balanced transfer has it's advantages, it also typically doubles the output voltage and finding an amp with 4 to 4.4 Volt sensitivity for full power output into it's balanced inputs could be quite challenging. Single ended RCA amps with a 2 to 2.2 Volt input sensitivity for full output is a much more likely find. Having such an arrangement will allow for the amp to output all the power you paid for, while minimizing the volume attenuation and the potential for bit stripping that can otherwise occur with such attenuation. As a plus it will give you greater range of the volume control and make volume attenuation less delicate and fussy.
 Your Classe' amp has an input sensitivity of 1 Volt for full output. While a high bit DAC will ordinarily have lots of extra bits to spend on attenuation without bit stripping into many amps; that 1 Volt input is pushing things a bit. It might work for you, but I offer no guarantees. 
 While not as ideal, using a passive volume control such as a LightSpeedAttenuator between your source and amp with short, low capacitance interconnects could work quite well in your circumstances.

I just sold a Gustard X26 Pro for $925 and it is an excellent DAC with decent digital volume. I hate DAC volume controls but the Gustard volume control was passable. I used a preamp with the Gustard. A used Benchmark DAC2 or DAC3 would also be a good choice with volume control. I have had both in the past (do not get the cheaper DAC1).

https://soundnews.net/sources/dacs/gustard-x26-pro-dac-review/

I am now using the Topping D90SE DAC ($900) in place of the Gustard X26 Pro ($1500). I was planning on getting the Musetec 005 DAC ($3500) after selling the Gustard, but now that I am listening a lot to the Topping D90SE DAC with the Topping pre90 preamp I am not sure I need another DAC.

I am also going to get rid of my Topping pre90 + Ext90 preamp soon. It is extremely quiet (wire with gain) and if you get the right amp match it is a steal. Unfortunately my amp does not match well with it. The Topping pre90 is ($600). It works great with my Benchmark AHB2 and Parasound A21+ amps but not the CODA #8 amp that it needs to work with.

https://soundnews.net/accessories/topping-pre90-ext90-review-pre-amp-my-power-amp/

So a used Topping pre90 and Topping D90SE combo can be had for under a $1000. If your warmish Classe amp is a match of the pre90 you will be impressed by this sound.

I am DM’ing someone who just bought the Benchmark LA4 preamp ($2500) and he will try it with his Class A Pass Labs amp. I also have the LA4 and it sounds EXACTLY like the pre90. The LA4 however mates well with all the amps I have hooked up to it and is not spartan in features like the Topping pre90.

BTW - the Topping remote controls are limited in distance covered.