@Marisa475 Well, you said that the host's stated objection to the lock was that she thinks it will somehow damage the door. Just because it doesn't say in the listing that you can't do something, doesn't mean you can.
A host can't anticipate everyything a guest might want to do and make some rule against it. That would be impossible. Am I supposed to make a rule saying guests can't kick my dog? And if I don't have a stated rule against it, guests should assume they can kick the dog?
I don't have rule that a guest can't repaint their room, but that doesn't mean they're at liberty to do so.
She probably never had anyone ask to put a portable lock on the door before. Maybe you could find a photo online of the type of lock you are thinking of, with illustrated instructions for installation, that would show her that it won't damage the door or the door frame?
And of course the host would also need to have a key for it, in case of emergency.
I assume you have booked this place somewhat long-term? If you can't work this out amicably with the host, I would suggest you proceed with getting the booking shortened.