I have a damaged book that I am sending free to someone.  She hasn't ordered any other books from my bookshelf. 

Is there any way to use the PBS postage to find out how much postage I need in order to mail it and to print a label with her name on it?

I can live without printing her name and address, but I'm at a loss for figuring out how much postage I need.

Yes- use Custom DC.  It's available from your account page.

You can post a dummy book to your shelf. If the unpostable is not WLed, you can post that one, but make sure to use a non-WLed book. Then you can use the mail to a friend button.

The custom DC will not calculate postage unless you enter in the weight and doesn't let you use the PBS postage.  It also requires the .43 purchase of DC, which you may not want to do on an unpostable. 

If you are sending the unpostable book to someone directly and it's not a wish-listed book, couldn't you simply use the unpostable book as the dummy book and use the mail to a friend button?