If you don't want plastic, I think felted wool would be your best bet. I'm in the process of making a felted wool quilt and it's been pretty easy--it could easily be adapted for your bed. There's many tutorials online.
First, a local thrift store has 50% off days about once every three months. I went to two of these and bought as many 100% wool sweaters as I could find. I prefer men's sweaters--they're bigger. I paid an average of $3 per sweater. It was in the summer so demand for wool sweaters was less than it is now.
Then, I took those sweaters home and washed them in the washing machine with hot water, high agitation, and high heat in the dryer. Then I repeated that wash process.
What resulted was beautiful soft felted wool. I cut the sweaters into squares--and am now stitching it all together. Felted wool won't fray or ravel, so you don't have to worry about that--just get it all attached to each other.
Plus--you probably don't need something big enough to cover both king mattresses. I bet you could get away with making something the size of one queen or one king bed.
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