You might want to buy another door (cheap and nasty or second hand?) and cut that one up as required to install the dog door. You just need to transfer the door handle/lock and possibly the hinges.
Then when you move, take the original door out of storage (where you kept it clean and dry etc.), remount it and swap over the door handle/lock.
This avoids the drama and possible cost when the owner says that he does not like the new door you are proposing etc. etc.
I could do it for you but am not back in CH until the end of August, you can PM me about it if you want.
But as you rent, you really do need to ask permission first to modify the door. If you go with the "I will just to do it and put it back the way it was when I leave" this may not sit well with the landlord and could sour the relationship.
It is not just about whether you are willing to pay for the change.
Of course had you asked for permission to install the dog door as a condition of the rental, you would have known how things stand.
Hope you find a good solution.
Even if they don't physically gain access through this, they may be able to reach keys.
We had a quote from CEBO for this item, including a dog door by dog mate but we felt the opening was going to be too small for our dogs but any larger the mesh would just be unstable.
So instead we decided to go with Morisibari, as in the photo below. It took a bit of getting used to but now the dogs don't mind them at all. We leave the door open only when we are home.
Dog doors have always made me nervous after my friend told me how she had a "visitor" who got into the house through the dog door and when she arrived home she found him and her dog on her dining room table.