Here is the tip my mom and I brainstormed out together.
Use fusible interfacing on the back of the pieces after you've sewn them together. It will stiffen the pieces up a bit, and flatten out the seams in the front.
You have to press a wet cloth to the back of the vinyl with the interfacing underneath it for 15 seconds. A few times, the vinyl seemed to warp under the heat, but generally, it's a much better finished effect overall. Make sure to do it on medium heat. And the "coated" side, incidentally, is the rougher side. That wasn't readily apparent to me the first time using this stuff.
Update on Harlan
2 days ago