I'd like to see a picture of the edge you are talking about.
As a longarmer, I never trimmed the customer's quilt. I left that for them to do.
Unless I was commissioned to do it, or to even finish the binding process as well as the quilting.
Anyway, let us see a picture of it if you can.
One thing you could do is cut narrow strips of batting and "couch" the batting strips along within the binding as you apply.