A great option for a CD player within your budget is to find a used Music Hall CD25, which usually runs $350-400, and upgrade the opamps, which would cost you somewhere between $50 and $150 depending on which ones you buy.
It's a simple do-it-yourself swap, about as complicated as changing a fuse. There are lots of other upgrades possible, from a number of different sources, but the opamp upgrade is very straightforward and makes a big difference in an already very, very good player.