If I understand your question correctly you want to know how to "cover" the edges of each applique piece. The bias tape idea would be lovely but time consuming. I'm always looking for the easy way out.
My suggestion would be to applique the pieces on your background and then use decorative stitches to edge each piece of the applique. (I just don't like working with bias tape.)
What a treasure you two are creating. I LOVE the design.