Great place - interesting mix of modern and traditional. Center a grid hanging of 5 over 5 (calendar art, botanical art, some theme) between the hanging chandeliers in the dining room on the back wall and it will work fine.
The green is a neutral and the brick red in it helps tie it all together. You need a neutral, and something mostly plain is best. For the rug, I think more muted - you can take a cue from the primary rug that has the wonderful sage tones with a little red. Flor is having a fall sale 25% off - I like the vintage vibe for the combo of earth tone and light but it is pricey at your size.

Of course, you can buy a 10 x 14 rug and just cheat the left end a bit - it would look good there. if you watch overstock you might find something you like - it usually takes about 6 weeks for just the right thing to come through at the right price. Here's an option you might not have thought of - the most cost effective thing is to go and pick out a carpet that you like and have them cut and bind the size you need. They will happily sell you the carpet at the per yard price and charge you nominally for binding. If you found a small print you liked, in neutral warm tones, it would work well for you here. You might have $ left for a rug for your seating area by the fire which should get the persimmon tone when you do it.