Can stackable Arduino headers be automatically soldered?
I would suggest using SMD headers on one side and through hole on the other side. One side with this: http://www.tracepartsonline.net/(S(30y0dp21y1rd0u3erv1mrrbw))/partdetails.aspx?Class=ASSMANN&amp;ClsID=/F_ASSMANN/ASSMANN.010/ASSMANN.010.170/&amp;PartID=10-18062010-079808
and one side with this: http://www.tracepartsonline.net/(S(30y0dp21y1rd0u3erv1mrrbw))/PartDetails.aspx?Class=ASSMANN&amp;clsid=/F_ASSMANN/ASSMANN.010/ASSMANN.010.170/&amp;ManID=ASSMANN&amp;PartFamilyID=10-23062010-067691&amp;PartID=10-23062010-067691&amp;SrchRsltId=1&amp;SrchRsltType=0

You will shift upper side a few mm from bottom side, and use glue to have upper side set in place and then reflow for soldering on both sides.