The sofa is a rather modern look to it! I would consider a wild print with all the colors of the room in the pattern. I also think a 9 x 12 might be a better choice as the room is very large and the larger carpet might be a better scale.

Here are suggestions for style of rug only.....you need to go rug shopping and find three that you think you like, bring home on approval, and then lay them out one on top of each other and see what you think....it truly can be trial and error. Just find rugs that have some of your room colors in them.....

Some caramel to blend with floor; very fluid

Feizy Bleecker Rug, Slate, 8'x11' · More Info

Geometric with shades of blue that would contrast with floor and relieve some of the caramel color in room

Bashian Dalton Aqua Area Rug, 7'x9' · More Info

Leather which give a nice texture and contrast to floor while blending with sofa

Bashian Aldrich Area Rug, White, 8'x10' · More Info

Very traditional; art deco sort of, but a nice counter point and the colors are really vibrant like a stained glass window; could work beautifully

Bashian Maribel Area Rug, Chocolate, 5.9'x8.9' · More Info

Modern; not sure if it reads grey or blue but a fun piece in your fun room.....purple pillows!?

Vintage Abstract Paisley Rug, Pink, 5'3"x7'7" · More Info

It’s adorable the way it is. Do not buy an area rug + leave the gold paint. Two shag white square pillows on sofa to repeat the flooring.

https://www.houzz.com/products/shag-michael-amini-pillow-decorative-pillow-white-polyester-filler-square-20-prvw-vr~41869912

[https://www.houzz.com/photos/sunroom-farmhouse-family-room-san-francisco-phvw-vp~40787[(https://www.houzz.com/photos/sunroom-farmhouse-family-room-san-francisco-phvw-vp~40787)

Thanks so much for comment. I did decide to leave the gold paint ... but did get a jute rug which I really like! Haven't done anything about pillows yet, and may follow your tip with white. Also considering a lighter neutral for the chairs down the road ...