I'd look for a quieter, mottled rug rather than a completely solid one: a medium blue-gray or denim ground with a little warm taupe would connect the blue chairs, caramel sofa and walnut without competing with the diamond-patterned chair. I'd skip a large ivory field with the black shedding dogs, and avoid another sharply defined geometric pattern. A subtle broken-up color mix would be my starting point.

For the practical side, shortlist low-pile options with cleaning instructions you can actually manage at the size you need. If you choose a washable rug, check the required washer capacity for that exact size before buying; 'washable' doesn't necessarily mean it fits your machine. I'd order a swatch or the smallest returnable size first and view it beside the sofa and chairs in daylight.

Before replacing the current one, lift it and check whether the lump is in the rug itself or a folded/deteriorated pad underneath. I wouldn't apply more heat without knowing its fiber and backing. What size is the current rug, and what budget would you like to stay within?

Current rug is 10x14 but I've been thinking of going less wide so there's no rug under the chair with ottoman but not sure. Would like to stay under $500.

Current rug doesn't have a rug pad. We removed the furniture and rolled it back on itself to see if that helped. It did a little but not entirely

That rules out a folded pad as the cause. For the new size, I'd mark out 9 × 12 before going smaller: compared with 10 × 14, it loses one foot across and two feet along the length, so it may pull away from more furniture than you expect. If the chair and ottoman are part of the conversation area, I'd try to keep at least the chair's front feet on the same rug as the sofa. If they're a separate reading spot, having both fully off the rug can look intentional; I'd avoid leaving an edge right through the chair-to-ottoman space.

With the $500 ceiling, I'd settle that footprint first and then compare actual prices for that size—small-size listing prices can be misleading. Leave room in the budget for a pad approved for your floor finish. A wide room photo taken in daylight would help judge whether the chair really reads as a separate area, but taping the two footprints will tell you much more than ordering a narrower rug on guesswork.

https://ruggable.com/products/ourea-dark-wood-tufted-rug?size=8x10&system=rug-cvr