I've never done that method, so I'm a bit useless for your specific question. Maybe the piece is too small? Or the opening?
What I do is get the Steam-a-Seam fusible web, put the not-cut-all-the-way piece of fabric onto one side of the web (it sticks!), cut the piece out exactly (with the fusible attached), put the piece and web onto the background, iron, and celebrate.
I'm actually having issues with stitching around it, since I've heard you're supposed to after it's all laid down. Put up a thread in the last hour or so about my problem, lol.