I like this rug, it isn't right next to the papered wall and keeps the space fun.

https://www.birchlane.com/rugs/pdp/paint-chip-blue-machine-washable-rug-b110018246.html?piid=87461233

I have two of those ( in another color) that Lyn posted. Bought mine off the Annie Selke site, check out their washable rugs, all 30% off today. Another possibility is Ruggable, I like their ”all in one” rugs, lots of choices.

For a (traditional Mediterranean feel), suggest going with a pattern with solid colored bedding.

Berber is pet friendly (had one in our family room for years…with an English Setter and a cat).

https://www.etsy.com/ca/listing/4399802295/boho-moroccan-wool-rug-beni-ourain?ls=r&external=1&ref=pla_similar_listing_top-1&pro=1&frs=1&content_source=a34fc3910a6777beda6ecf36936b0e34%253ALTcfbd3e98316d5bb33889fa97f34618c320f9d79c&logging_key=a34fc3910a6777beda6ecf36936b0e34%3ALTcfbd3e98316d5bb33889fa97f34618c320f9d79c

If not comfortable with mixing and for a paired down feel, suggest a plain rug but be sure to add some pattern on the bed. Fyi: We had jute and sisal rugs and no issues with pets.

I woukd look at a natural woven rug The wallpaper is very patterned and busy .. its the focus

Jute or seagrass would work very nicely.

I see nothing mediterranean about the wall paper and it is IMO very busy so what drove that choice I guess I would want to know. The molding already busy IMO anice blue wall color and then a simple choice for a rug with a more Med feel and why pet friendly ? That says it needs to be washable and FYI once a dog or cat pees on a rug the smell is always there for the next animal to think it is okay to pee there.