If you have some experience at FMQ then learning to use a ruler foot and rulers is not much of a learning curve. Using a circle ruler and a ruler foot you can do circles without having to rotate your quilt. I've just finished doing a small wall hanging with quilted circles that were done using a westalee ruler foot and one of their circle rulers.
If you are using one of those circle tools like the one that Nancy Zieman sells, then those are designed to be used with the feed dogs up and a regular presser foot, though a walking foot might work as well. Anyway, the feed dogs move the quilt and the circular tool has a pin attachment that makes sure your fabric is rotating around a fixed point. It seems to me that unless you're doing a tiny quilt then as the quilt rotates it could very easily get bound up against the tower to the right of the machine and would begin to twist. That would cause puckering no matter how well you've basted your quilt. You could try sewing very slowly just a stitch or two at a time and carefully checking each time you've had to move the quilt to make sure that there are no puckers forming under the quilt as it rotates.
Rob