The best and most efficient way of installing a doggy door is to buy and install the kind already made to go into the door.
Cannot understand your obejction to such types.
That said and what I use is a storm door with a doggy door already built in.
Works well in summer and winter.
I just leave the entry door ajar and the dog and cat both know enough to open it an let themselves out through the storm-pet door:
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