If you are unable to manage the curved seam by holding the top layer up off the other piece and only lowering it as it meets the needle, there is a product called the Curve Master Sewing foot that might be useful for you. Make sure you buy the tweezers and wooden roller to help you master the technique more easily. You really need to have a more experienced sewer show you the technique for sewing curves and then the process is easy.
Another slower method is to pin at each end of the seam and then ease and pin all along the seam. Some people even gather very slightly when easing in and then clip the curves.Try not to stretch the bias edges. Good luck with it all.
P.S. As for the square: You could always applique it on and then cut out from the back in order to make the thickness consistent.