stick framing scissor trusses
So you have a 1.5" gap between each ceiling joist pair? If I'm envisioning this correctly, here's a suggestion, the other guys will chime in:
Install perlings in between the rakes of the rafters and the ceiling joists, located such that you can install a collar tie between each set of ceiling joists (in the gap), either at the apex or down a little, so you have a flat spot at the top of the cathedral. The collar ties will coincide with the rafters if they're in the gap of the ceiling joists, so they'll have to be anchored to the perlings very well. Connecting the ceiling joists to the collar ties, which are connected to the perlings would (in my opinion) form the equivalent of a truss setting on top of two scissor trusses, which should be much stronger than what you've got and you wouldn't have to tear anything down.
Just a thought...