I don't use interfacing. I just needle turn mine.
I use a compass (or computer) to make the circle the size I want. Then, I cut it out of a cereal box. I trace it onto the wrong size of my fabric & then add 1/4-1/2" before cutting (seam allowance doesn't have to be exact -- 1/4" is good for hand applique; I'd use closer to 1/2" for machine applique). I hand stitch a running stitch halfway between the cut edge & the drawn line. Next, I set the template back onto the wrong side of the fabric & pull the thread so it closes around the template shape. I spray some starch/sizing into a small cup (or the lid) and use a Q-tip to paint it onto the gathered edge of the circle, making sure to smooth out any ripples along the turned edge. I press the painted, gathered edge until it dries. I gently loosen the fabric a bit to be able to remove my template. Then, I pull the thread so it gathers back up again & give it one last press back and front. Now it is ready to be stitched down to my background fabric.
I typically don't need a stabilizer, but if it's still gathering it like that, I'd go with Pellon tear-away paper stabilizer. It goes behind your background fabric.
Cute fabrics!