Hi Chris,
Have you tried adjusting for distortion and vignetting in software before combining the images? For me it can help quite a bit before attempting to stitch them together. Also if you have plenty of overlap (i.e. using say 4 panels instead of 2 to cover the same area) then you should be dealing with the less distorted central parts of each image.
One more suggestion (and I know its obvious so please don't take offense) but wait for a completely still night. Much easier to deal with tree movement if their isn't any

. Given that it is your back yard, you could also take the foreground image on dusk with low ISO and fast-ish exposure, no stacking required, no movement issues, then blend that one with your stacked milky way pictures taken later that night.
Cheers,
Tim.