You can avoid losing Pet Pal.
1) Simply refuse when she offers to teach you more source. If you already have 3 source points, you don't need any for Hanag and this also saves 3 bears lives. Hanag will still give you a source skill book.
2) Mirror Mirror in the Lady Vengeance. There is a way point right by Hanag. It only adds about 2 minutes to your game time to pop back to the Lady Vengeance and remove the Talent. Then go do Hanag and let her kill the three bears. Not sure if Alice is suppose to be Goldylocks.

Anyhow, after doing everything with Hanag just go back to the mirror and get Pet Pal talent back.