What I would do is cut the size of the fiberfill you want, make the covering ~ 1" larger then fold the covering and stich through both top and bottom to stablise the batting between two layer of fabric so it can not shift in the wash, could also quilt it if you can get your machine to pull a tight enough stich or would not be a chew toy or get caught on pigs nails, not sure how good the moderen machines are at that, I am used to the old cast iron Singers you could actualy adjust well.
I am by far not a seamstress but have done a lot of sewing on things like copied motorcycle seats and car seats taking note as I take them apart how they were lapped and stiched for strength. 99% of the time durability depends on at least 3 layers of fabric stiched or it will pull out, look at levies and note there are 4-5 layers on the main leg seams & double stiched.
Kristin
* Got to get me another old singer machine soon, lol I have even used a treadle from time to time, talk about 'old school'