This is for the OP.
I have though about this off and on without judging the quality of what you want to do.
Cut the darned hole however you can. Place the unit in place but do not secure it yet. Cut 4 rectangles of lightweight plywood, I would use 1/4" Luan as it rots quicker, that will cover the gaps on the outside of the building and not block the operation of the fan. Drill a hole or two in each depending on the size and dangle them on the side of the using a heavy string or wire. Secure the fan. Pull the panels into place to form and exterior form for your mortar. Secure them to something inside. You may want to have a inside piece of plywood that covers most of the void on the inside.
Then use a grout bag to fill the void with mortar. You will not be able to do it all in one shot, but over a period of a few days you can fill in the voids. Cut off the string or wire on the inside and finish the wall. The exterior portion will rot and fall off the building in 1-5 years depending on the climate.