I don't think that making the rudder axis vertical will add much efficiency; it will, however, add some problems of making the structure sound, as you guess.
I wish I had a skeg. Please don't cut yours off, but extend it aft to the transom at the same depth.
Then bond a parallel sided piece of wood, not triangular, to the transom (your 48mm sq spruce will do, but I'd rather use something harder), long enough to bond to the after end of the extended skeg, so that they give each other mutual support.
Cut away the balance area of the rudder in way of the skeg. Then you can use Annie's figure 8 lashings at deck level and at the bottom of the skeg, for the very best frictional performance (your vane gear will appreciate this).
The draught of the finished rudder should be at least 20cm less than the draught of the keel. You'll be thankful for this when you bounce the boat on a hard bottom.