There is a trick to working with so much bias and seams that intersect like this. Number one.... Never scrub iron the seams... Always press and lift and press your seams. Number two, press your seams rotating in the same direction. This distributes the bulk around so the fabric has room to lay flat. I have heard of some women using an eighth inch seam to help with the bulk too.
I have also heard to begin with fabric that is highly starched, that gives you a little lead way helper with all the other things I have mentioned.