A forest green would be better.

I like your purple couch:) I think the green rug could work...I like that it has two different greens.

Your rug reminds me of a purple/green rug I adore but not brave enough to buy

I was also going to suggest a sisal. Or as a_p_o suggested, forest green.

What about having a piece of carpet bound?

It might be easier to find just the right color and larger sizes.

You have some gorgeous pieces! I like them together and I think you could accomplish a great eclectic space with the right rug and drapes to ground it.

Check out Kim Parker for ideas on how to own this style:

https://www.houzz.com/projects/33649/kim-parker-interiors

None of these individual pieces "match" but look how great they are together when united by a common colour, in this case the blue:

Willow Glen Residence · More Info

In your case it could be the deep burgundy/plum (hard to tell the exact shade on my screen) or forest green.

This has potential.

Tibet Rug Company - 60 CT Maya - 8ft 0in x 10ft 0in Forest Green · More Info

I hear you on the sheers. I feel the same. What if you replaced them with bamboo blinds? You would never have to roll them down and they would add some texture.

Bianco Roll up Blinds, (48 in.x72 in.) · More Info

Rats!

If you did a different layout you could probably get away with 5x7 or 6x8 (cheaper). Could you float the sofa in front of the fireplace, facing the two armchairs? And move the side chairs to the bay window? This way only the fronts of the furniture would have to be on the rug and the smaller chairs could be off it completely.

You need to go with a neutral rug and add the deep green and neutrals that is in the sofa in pillows and maybe cover your seat cushions.