You can't draw a button across two panels: what you draw on a panel stays within the bounds of that panel, and will be clipped outside it.
What you can do is not draw the button on either panel, but on the underlying form. Then either set the z-order (using the "bring to front" and "send to back" buttons in the designer) or with the
Control.BringToFront[
^] and
Control.SendToBack[
^] methods at runtime.