Few things here:
1) If you click directly on the rat from a distance, your character will run up to it and open a dialogue the moment it gets into range. This is easier than trying to click near the rat's current location, and then clicking on them again to attempt to chat.
2) More importantly - the rats aren't running from your character with pet pal. They're running from your characters *without* pet pal, who are probably chained to your animal-loving character and following them around everywhere, which in turn is causing the rats to flee from them (and as a result, you). Whenever you see an animal you want to talk with you'll have a much better result by disengaging from the party (sever the link by the portraits) and walk calmly over to the creature - it won't run as long as the rest of the group keeps its distance. Once you've got it talking you can safely bring the rest of your party over.