Damages cause by Dog, should owner reimburse?

You're really quite fortunate that the owner offered to reimburse. Definitely take it.

The alternative--viewing this as a cost of doing business--is however very accurate. I expect that, in the future, you'll take whatever steps are necessary to ensure this doesn't happen again. Guest dogs shouldn't have free access to all areas of your home, especially when you're not there. Since the dog scratched up a door, closing the doors to rooms you don't want them to enter wouldn't seem to be a solution. You may need to crate dogs who exhibit destructive behavior overnight and whenever you leave even for short periods of time.

Absolutely not the owner's responsibility. The dog was left unsupervised in your home, and not contained. It is your responsibility. Expensive lesson to learn. Crate dogs when you aren't watching them.