I’ve owned quite a few integrateds within your budget. The standouts are Yamaha’s MOSFET powered series (A-S2100 or 2200), the Rogue Audio Cronus Magnum 2, and the Axxess Forté1 (within your budget used). Most others were good to mediocre but did not connect me with the music. I like Rogue’s full tube products but I would avoid their hybrid amps. I’ve owned both the Sphinx V2 and Pharaoh. Both were mediocre in my system and left me cold.
The Axxess Forte1 is my current integrated. It doesn’t compete with $20K of separates (hardly any integrateds do) but it does remarkably well and most importantly, draws my attention toward the music rather than having me critique its performance every time I sit down. A bonus is you can leave it powered on 24/7 with negligible power draw. I preferred this integrated to a Luxman 507Z in a back-to-back comparison.
The Rogue CM2—great sounding amp—engaging, but was noisy in my system. Allegedly they addressed the noise floor in their mk3 revision. If so, I think it is a no-brainer if you can live with the heat of tubes. One thing the CM does particularly well is throw an extremely wide soundstage—on par with megabuck amps. To my ears it only sounds solid-state in the good ways but offers plenty of tube bloom and dimensionality.
I’ve also liked Rega integrateds throughout multiple auditions. They tend to have excellent PRaT.