The navy is already well established in the chairs, so I wouldn't make the new rug predominantly blue. I'd look for a warm flax/tan or oatmeal ground with a little softened navy and gray in the pattern. That would connect the chairs to the silver sofas while still relating to the warm wood around the fireplace. Your current rug's quiet, varied pattern works with the room; a larger rug in that same spirit may be easier to live with than a dramatic new color.

For this main room, I'd use your concern about the wool/viscose blend as a useful filter rather than talk yourself into a rug you already worry about. Compare 100% wool options in a low or medium pile, then check each maker's care instructions and return costs for the 10x14 size. Lay the samples on the floor beside the sofa and chairs in daylight and at night, not just against the upholstery in your hand. Before ordering, tape out 10x14 to see where the chair and sofa legs land and whether the rug leaves enough wood showing around the fireplace. Of the samples you've tried, is there one whose color you liked but whose fiber or pattern ruled it out?

I'd choose something with a bit more statement value to tie in the fireplace.

https://www.perigold.com/rugs/pdp/madison-couture-rugs-tribal-elegance-bluerose-hand-knotted-wool-area-rug-p100220980.html?piid=907527142

Add a larger end table with a table lamp where the two sofas meet, or swap the navy chairs and the loveseat if you need that spot open.

Replace all the navy accents...just way too much of it. Choose a rug which works with the gray sofas, works with the two navy chairs, and invites the warm tones of the floor and fireplace trim into the mix.

https://www.perigold.com/rugs/pdp/madison-couture-rugs-wool-area-rug-p110033832.html?piid=1893700445

When you get the new rug position the furniture closer and purchase a much larger coffee table. I like this one from Wayfair because it acknowledges the stain of the floors and the fireplace.

Next time you purchase a sofa buy two; not a sofa and loveseat.

Viscose is NOT CLEANABLE. The minute water hits it, it turns brown.

I'd fix that chair you've placed in a corner to make it a more desirable location for reading.

The dark navy blue pillows could remain as a backdrop for patterned pillows.