WOW! This is a quandry, for the very reasons you stated...
My thought would be to use a really really narrow zig zag stitch, just zigging back and forth enough to be seen. Then you could quilt your lines going straight across, and the little zigs would hide if you were slightly off on the corners. Nancy Zieman calls it a "wobble stitch."
I use it for SID, because it is forgiving...
Maybe try it on a scrap to see if you like it?
I really like this quilting design on this pattern :D:D:D