I'd just use a heatpump/inverter.

There could be so many reasons why that room is doing what it's doing. Is it north facing/getting all the sun? If so are the roof eaves far out enough to stop the summer sun getting in, yet receeded enough to let the winter sun in? Is the giant concrete slab in the garage floor acting as a heat/cold reservoir in the summer/winter (the thermal content/capacity of these things will mean nothing you can do will affect what it wants to do for a long time - it's got inertia. Do you have a good heat source heating the rest of the house in winter that this room misses out on?

Just install an inverter heat pump.