I know I am not answering your question, but are there no other options for the pet door? (like putting it in a patio door instead?)

It just seems like a difficult and cost prohibitive thing to “undo” when you no longer need it and/or go to sell the house.

We installed a storm door with the pet door built in. We love it. No cutting holes in your door, plus it's easy to block by just shutting the inner door. Better for security as well as keeping the dog inside when you need to.

Purchased at Lowe's. Took an hour or so to install with a drill and a few basic tools. The only part that's a bit of a pain is you have to drill holes in the storm door to install the handle. But there's a good reason for it, so you can position the storm door handle so it does not get in the way of your regular door handle.

Here's a picture from the web, this is not our actual house.