You dismissed attic ventilation as not being an option, but I think you need to know it may not be as difficult as you think.
Basic roof venting is comprised of both intake and exhaust vents. The ratio is 1:1, 50% intake and 50% exhaust. Intake happens down at your eves, and through convection the heat rises and goes out your exhaust vents near the ridge (top) of your roof. Most of the time, the problem with ice dams comes from the lack of intake vents. The most common intake is vented aluminum/steel/vinyl soffit, followed by wood soffit with holes drilled in it and covered with vent grills. Adding these intake vents at ice dam areas is the typical solution, but there are more solutions that you probably don't know about.
If the exhaust net free area is too small, then you can add a shingle over ridge vent. If you have a hip roof and don't have much horizontal ridge lines, you can also use hip vent. Alternative solutions for adding exhaust vents is with Omni-Wall vent for where the roof meets the wall and you can't put a regular vent. Adding regular roof louvers is the obvious option, but like I said earlier, exhaust is rarely the problem with ice dams.
Alternative solutions for adding intake vents in places other than at the soffit are vented drip edge and shingle-over edge vent. Either of these options should work.