First why a fake rug get a real one you don’t fool anyone with a fake. As for bedding go white it is classic looks great and a no brainer for the rug..

more info about the room and your style preferences would be helpful

I usually use plain comforters/blankets and then put a duvet on top. So I can variate the degree of warmness as needed etc. That depends on climate, personal preferences, etc. Of couse you spend then on duvet(multipe duvets in my case 'cause I like to mix up stuff)-but you're less stuck with one kind of coverlet, regardless seasons, your mood, etc.

It was a (long) time in my life though when I had one patterned comforter, so let us know whether it's something you'd prefer.

Otherwise..say you pick one by its qualities..warm, cool , thin, thick..then dress it wit a duvet..how about linen? Very comfy, relaxed enough to work with the rug..and you can use throws/pillows to tie it more in terms of colors/play with it if you wish

Just as an example..

https://www.garnethill.com/solid-relaxed-linen-bedding-and-pillow-cover/bedding-home/duvet-comforter-covers/498717?listIndex=1

Frankly, all the colors would work with the rug as long as you decide on your direction.

(I think they were having 30% off recently, the site)

Now, there are also coverlets and quilts..thinner than comforters, usually, but thicker than blankets..did you mean something like them?