I'd probably applique first but that's me cause I like to quilt around the edges so that the applique pops. But I do hear you not wanting to have tons of stops/starts. One way that you might handle that is to actually do couple of back stitches when you come to the applique, lift needle and then "skip" over the applique and put needle down and do a couple of small stitches and proceed with quilting---then you can trim the loop of thread left over the applique after you finished. Pretty cool quilt!