Placement on top of a dresser or even on top of a bookcase, will yield much better results than inside a bookcase. Inside a bookcase leads to a box inside a box situation and that is just tough to work around.
I have had speakers placed in this manner for most of my life (65) and have found that you don't have to go crazy price wise, so your budget is more than enough.
Just realize that the goal is to get things to sound good. Some of the audiophile niceties will get lost with this type of set-up, but that is ok.
Of current production speakers, I have had the most success with:
Omega Super 7MKII - $1K -(have also had the Super 3i & CAMs. The Super 7s are preferable. Used with a Rega Brio integrated)
GoldenEar BRX - $1700 - (currently in use with a Rogue Sphinx integrated). The Aon models could satisfy the look factor you want.
In a small bedroom you can also get creative with your choice of amp. You don't need tons of power. Something like a Jolida Glass FX 10 or even a vintage receiver (thinking something like a Marantz 2216B) could look very cool.
Good luck.