Budget integrated amp needed

Buy a GT102 from Dan Joffe at Akitika. You can get one for about your budget with optimization for either 4 or 8 Ohms. I can’t think of a good reason to not get the Z4 for versatility. This is an easy kit to build if you have some spare time. The Node is a preamp/ DAC/ streamer/ HPA in one so power is all you need. 

I personally wouldn't dump a lot of money into a bedroom system.  What about a living room?

Th VR-2020s are a 10" 3-way with a 91dB sensitivity, and are a relatively easy load to drive. And while you could go off into any number of esoteric rabbit holes, your budget is best served by a high-value integrated amp. And right now the King of that Hill is the Marantz PM-7000N. At 60 w/ch, 80 into 4 ohms, you will have plenty of power. And you will have the full connectivity from a very good phono stage, 3 more analog inputs,  to built-in 24-bit/192kHz PCM streaming across the digital inputs (USB, coaxial and two optical) and music servers, and 5.6MHz DSD streaming across the USB input and music servers. Plus the HEOS remote. I stream 256K AAC regularly, and I'm very impressed. Dont just believe me, read the Absolute Sound ("50 Greatest Bargains In High-End Audio"), What Hi-Fi, and other reviews. Sure, there are better integrated amps, but they're all multiples of the cost - the Hegel H95 is often compared, but it's $2000 and lacks all the flexibility of the PM-7000N. Luxman, Mcintosh, and Rogue Audio (Tubes) all make great, if very analog-centric amps in the $3-5,000 range. There's always used, with all the attendant benefits of cost, offset by lack of warranty, and lack of current technology. If you were interested in going vintage, then that's another discussion entirely. But for your budget, nothing will get yo farther down the road than a PM-7000N. 

I +whatever the Yamaha A-s301. The NAD 316BEE can be found used at your budget. I am familiar with both and can recommend.

I have an Audioengine N22 ($199) that works very nicely in my bedroom system. Unlike most amps in this price range, it is NOT a class D amp but rather an A/B design. It is not as powerful as many others at this price point, but if you have reasonably efficient speakers, a smaller room or simply don't listen at higher volume, it may well be more than enough. I certainly enjoy its sound. (They even have refurbed units for $159.) They have a 30 day no-questions full refund return policy. No restock fee, just the shipping cost.