I really don't know if this would apply to what you want to do with the quilt, but......I would handstitch the applique. Then do the quilting.
I quilted a quilt that had been hand tied. This was a lady's first quilt and she hand tied it to get it finished quicker. Well, something got spilled on it and my daughter (it was her mother in law's quilt) asked me if I could quilt it for her. At first I just thought I would do the meandering in big areas on it to keep it together and then I found out that she had the cotton batting in it. So what I thought would be a more simple job, became something that took alot more time to do...
But quilting it when it was done was not hard to do. As a matter of fact, it was a little easier. I have a longarm so it wasn't that hard to do. But it turned out nice in the end.
Before I got my longarm, I sent my quilts out to be done. I worked so hard to find just the right pattern for my son. He loved it when it was finished, but after about 1 1/2 yrs the stitches came out from the quilting. So I asked him if he wanted me to requilt it. As of yet I haven't gotten it from him, but as long as I worked on the quilt itself, I don't want him to just get rid of it because of the finished quilting that has come out.
I might call him today about it. Thanks for reminding me of it.
Barb