How about getting panels that you use to block in a window air unit? Raise the window, bolt the doggy door to the bottom of the lower sash that you raised and then use the air conditioner panels to block in the side(s).
You could probably get the panels still this time of year from Lowes/Home Depot, I'd think...
An alternative that would maybe not look quite as nice but would work would be to use plywood to fill in the area to the side of the door - build a little frame that would be the size of the window opening out of 1x2 or 1x3 and then cover with plywood.
Just a thought - not personal experience here.
Susan