The menu is painted as part of the non-client area. So if you are doing your own non-client painting, you'll also have to draw the window yourself. You may be able to hack something out with TrackPopupMenu to do the actual menu drawing, and you'll just have to handle the menu bar and top-level menu items yourself.
Drawing the non-client area yourself is fraught with peril. Are you sure you don't just want to use an owner drawn menu?
It can be tempting to tweak your UI dialogs to fit your exact needs, but also keep in mind that it is jarring to users who are accustomed to the look-and-feel of windows already.