Nice space! I understand your concerns about the floors and the budget but bigger is better. Furniture arrangements look best when all the furniture in a grouping relate to the rug.

eSale rugs often has reasonably priced rugs and they have free shipping and returns. If available, get the smallest version of the rug you are considering. You can check out the weave, colors, etc in your space and if it doesn't work, you don't have to lug a giant rug to the post office.

I'd love to see some colorful pillows on your sectional too.

You can go 2-4 feet wider than the sofa. Rug should be big enough to slip under the front 1/3 of the seats and extend past the side of the chaise and the other end of the sofa by 1-2 feet. Leggy chairs (which yours are) should sit entirely on the rug. Use painter's tape on the floor to tape this out. If the taped "rug" is not a standard size, then go up to the next size that is standard, off the shelf.

thank you both for your input. I am a bit sad at covering thousands of dollars of custom hardwood, but it sounds like I need to do it.
I like the idea of painters tape to mark out the area.
Not sure I can get a standard rug as big as what I need, but I will certainly look.

It will look really nice, and warm up the room, bringing some more life to it. It's a lot of fun to choose a great rug! You will also need a good quality rug pad, preferably the felt kind, so nothing about the bottom of the rug really rests on your wood floors. Protects the floors and also makes the rug feel cushy underfoot.

With our kids and dogs, we also keep a water-proof rug by our back door, to protect the floor by the entry doors. But then, we are heavy in and out door users.

Just be aware that your floors will fade with sunlight, so whatever is blocking the light, such as furniture, rugs, etc, are areas that will stay darker under those furnishings.

Life and scratches happen. I have 2 dogs and have learned to love my scratches as a reminder of the joy that the dogs bring! Embrace the patina. If your floors are perfect in 10 years, then you've been worrying about the wrong things!

JMO, but since it's what you'd like, I wouldn't hesitate to put a rug under only the couch/ottoman area... It would protect the most-used section of the floor and leave the majority of floor exposed with a clear path to the right patio door.

Make sure you get a good pad to go under that rug. I am currently in the process of repairing the wood floor in my family room as I didn’t have a good pad under the rug! From recommendations here I ordered pads from Rugpadsusa

I am not sure if your furniture is make do or things that you love and want to keep. It’s all interesting and nice but I really think you need a larger rug to tie everything together. The chairs already look a bit orphaned. If you leave them off the rug it will just emphasize the issue.