There is no way to sew it on with a machine. You could double stitch the lip, as you call it,and trim them down or you can trim and overcast. This is so it won't revel. Put the seam on the bottom of the cord next to the pillow and sew it on by hand. I would lay on the cording in your design first to give you an idea of how it would look.

Depending on how thick the cording is you may or may not like the look and there's not much sense doing all the work and then not like it.

If it doesn't work out, there are many ways to embelish a pillow from sewing on buttons to sewing on doilies. I'm sure you can find lots more ideas on line

Thanks oilpainter. I've kept thinking about it and I still like my idea, but it does seem like there would have to be a better way! Think I'll go prowl the trim section.

Agree with Myrna on the raw edges posing a problem, unless you put a covered button in center to cover the raw edges..
Do you want to use it like say a decoration say like a frog? That is the only thing I could think of that resembles the knotted couture look on some pillows.
What if you made a fabric tube incasing the cording?
There would be no raw edge. Below is a site for the frog but maybe you could use a version of this.

Here is a link that might be useful: Frog Instructions to make knotted decoration

Thank you Patti! I was thinking of that also, and thanks for the link. Don't know if I will try it or not, but if I do & it works, will let you know. Merry Christmas!

Sew it on however you want, but not close to the cording, closer to the edge. Then turn it over the stitching and stitch again close to the cord. There'll be a seam that shows, but it will be finished and part of the design, like a double welting, one large and one small.

You can sew it into the pillows, but you have to make your own pattern and add seam allowances wherever you cut the pillow to insert the welt.

You can make a covered cord with the seam allowances inside. Then you can sew it onto the pillow top by hand. I suspect the easiest way would be to very carefully pin in place, and sew on from the wrong side.