Try a mid-century modern rug like Hemphill's Rex Ray Area Rug Collection or Jaipur Traverse Caracas. I looked up 'mid-century modern rug' and chanced upon these on Houzz. I like the muted colours which would soften the cream on brown. The mid-century motifs would highlight the mid-century chair. Good luck!

With the soft green walls and heritage character, I’d keep the room warm and classic rather than too modern. A medium warm timber stain would suit the floor well, while off-white, linen or light beige roller blinds would keep the space bright. For furniture, I’d mix natural or walnut timber with neutral pieces, such as a beige or soft grey couch and a textured rug in cream, muted green or warm grey. I’d also add a simple mantel over the fireplace even if it won’t be used, because it will help preserve the heritage feel and frame the large canvas nicely. A few plants, warm lamps, framed pieces and subtle brass or timber accents would also bring everything together without taking away from the original character of the room.

I have a similar family room, with big windows and a sliding door and we have block out Roman blinds on them. They sit at the top of the windows most of the time as we don't have any privacy issues with being looked in at by the neighbours. We really only put them down at night in the winter. Ours have a creamy coloured background with taupy wavy lines on them. Very neutral. I never could have imagined that's what I wanted til I saw the fabric. Good luck I'll see if I can find a photo to post.

Suggestions from neutral colours: https://www.dulux.com.au/colour/colour-trends/2018/essential Ceiling & walls the same colour is a good choice for square set. There will be subtle hue difference in various lighting situations anyway, which provides depth and makes for an interesting look.