Killer used integrated for

I've been looking for something similar. Mine is specifically for a pair of EPOS M5 speakers which are 88db, 4 ohm bookshelf speakers. (Recommendations welcome.)

A few things that have been on my radar are the PS Audio Sprout, the NAD 3020, the Emotiva TA-100, and the Nuprime iDA6 which is on clearance for $350. You can probably get to some things a bit higher than this list if you are looking used, but of course that's dependent on what you actually find available used. 

https://www.usaudiomart.com/details/649634586-vincent-audio-sv129/ $385 used, down from $1k. 

I found an older Audiolab 8000A for $300 a few years back and it's great. 

For that price used, try it and move it on if you don't fall in love.

Just keep what you have, swap the amps to 2 channel and be good. Not going to find much better for the $ you are talking about. And you get Audyssey to deal with room funkiness.