Advice requested for making a vinyl quilt rug Sewing Machines Parts and Sales in Los Angeles

I’m giving up on store-bought area rugs since my four dogs make it a losing battle. I have an idea to make my own area rug in a quilt format using vinyl squares and batting. When I work it out in my head, the most layers of vinyl after hems and joining would be 16 at the corners. That sounds pretty thick. My machine is a 20 year old regular Kenmore so I’m not sure if it would work. I never changed the feet before. Maybe I should only do vinyl on one side and have fabric on the side facing the floor? That way if the dogs make a mess it would still be on the vinyl side. I’m just brainstorming here and any ideas y’all have are welcome.