Will they let you mount conduit onto the outside wall if you can do so conspicuously? PVC conduit can be painted if you scuff it up first.
We regularly mount transformers and irrigation controllers inside finished garages. To run the wires through the walls we simply drill through the wall and slide in a piece of PVC conduit, usually 1.5" or 2" depending on the number of wires. Be sure to insulate around the pipe if it is an insulated wall. We then use a junction box on the ends of the conduit on each side of the wall. That gives a nice finished look that covers the pipe in the wall. On the outside we then run a piece of conduit from the junction box to just below grade. We then use silicone to plug the end of the conduit underground so that no creepy crawlers make their way up into the conduit. On the inside you can have the wires come out of the inside junction box or for a more finished look, we'll run conduit from the junction box to the transformer.
We paint the outdoor PVC conduit, junction box, and fittings/mounts to match the house. Last thing we will do is put a piece of insulation in both the inside and outside j-boxes to eliminate heat and cool transfer.
Why do people not respect us as they do other tradesmen? Because every Tom, Dick, and Harry doesn't think he can be a plumber or electrician!