$1,000 to spend on room treatment....

Beheme, that's good news. I would suggest also focusing on the parimeter of the front half of room that is not covered by the thick rug. Decorative pillows can help.

Regarding the bump in the 200-250 Hz region and to further improve the sonics, I would suggest you continue experimenting every few days with speaker placement. Eventually you'll find the best location and/or you'll eventually give up realizing you can't improve much more with regard to speaker location and the speaker's interaction with the room.

Thanks for trying some of my suggestions and despite your room's issues I remain committed that the room's acoustics have far less influence on the sonics than most are willing to admit. But that commitment is obviously qualified with the assumption that a given room has certain basics already addressed such as wall-to-wall carpeting.

-IMO

My experience has told me to go natural. Natural fibers natural jute back pad under the carpet instead of the foam rebond products normally supplied. The jute has more area of absorption and less bounce back. Also find that natural long hair sheeps wool used in the construction of sound panels makes for a more linear and musical result than does foam or fiberglass which tend to suck out and reduce dynamic contrasts..Tom

I fixed a flutter echo from hell by placing an oriental screen in the back of my room. You could probably get a nice one used for not much money.