Integrated with balanced inputs under $1K?

Cambridge Audio 851a
$1500 new you could probably find a used unit for 1 Large
https://www.cambridgeaudio.com/hkg/en/products/851/851a

one here in old blimey for $864.22 usd

https://www.ebay.com/itm/Cambridge-Audio-Azur-851a-Black-Stereo-Amplifier/222978799261?epid=15062415...

Pathos Classic One MKIII

I recently (last summer() purchased a Pathos Classic One MK2 here on AG. They can be found for under $1000. The Classic One is a hybrid design that delivers 70wpc into 8 ohms, 130 into 4 ohms. It has a single balanced input, is a hybrid design. A beautiful piece having a wonderful sound. How demanding are your speakers regarding load on an amplifier?  

@mesch The ASW's are 90db at 2.8V with a 4 ohm imp.

Look into the Schiit Ragnarok...above your budget but their devices are well made.

Creek Audio Evolution 50a.  New just abit over your budget.  Used no problem, in fact there is one for $600 listed here.