I can not recommend a particular single box player. But I can give you some impressions about a separate DAC. The separate DAC basically turns the digital information on the CD to an analog signal to be amplified. It replaces the DAC that is already in the CD player. I think the idea is that as a separate component, the DAC can be more sophisticated and deliver better sound .
I use an MSB Gold DAC modded by Empirical audio. Other good ones in your price range are Benchmark, Perpetual Tech,Scott Nixon,Ack!Dac! You can read about them and single box players in the Digital Domain circle.
Now for the main point. My system is much much better with the separate DAC. The music is more dynamic, less digital sounding, clearer than through the CD player alone. I use a Phillips 935SA as a transport. Its below your price point, but still a good player. When our DAC was being modded by Empirical Audio, we listened to it alone. It actually sounded pretty good, but a little thin and uninvolving. We listened to less music those two weeks!
I would recommend you try a separate DAC,