The last quilt I made, I laid the back on the floor and put batting pieces over it to cover. (Lots of pieces.) Then I moved each piece, put Elmer's School Glue onto the back, and put the piece back down. They were all overlapped about 1/2 inch. After everything was down, I used regular fabric glue, the kind that does not wash out, and put a drop every few inches under each overlap. That way I knew the batt would not move and shift if I hadn't quilted closely enough. The quilt had to stay on the floor for a day (bummer) so that the fabric glue could dry in place.
When I FMQed it, I could feel the layers with my fingers, especially where more than two layers came together on corners, but normally you would never notice.
Small, really scrap pieces of batt do not go into dog beds here. Only fabric scraps (the lint) go there. I have a small wastebasket lined with strong fabric sewn into a pillowcase-like shape, and all the batting scraps go into it. I try to pull the batt pieces apart so they are more fluffy. Then when the basket is full, it is sewn shut, and the resulting "pillow" can be donated to the food pantry or Salvation Army etc. I try to get them stuffed quite firmly. I've made dozens of these. Our food pantry lady loves them.