I am baffled by some of the posts on here. I feel sorry for all concerned here. I think it's a very difficult situation to do the best thing for everyone, actually.
Yes, it sounds like the step-daughter is struggling and she certainly needs support. This does need addressed, but does not somehow negate the fact that what she has done is completely unacceptable. It was designed to hurt her step-sister (and step-mother, presumably) and was extremely nasty. It could lead to serious consequences, although I can see she probably will not have fully grasped the potential consequences- but she certainly knew what she was doing was wrong. I fail to see how giving her treats/holidays/fun activities is going to help matters. She needs firm boundaries in place, not buying off or rewarding bad behaviour with treats. Putting those boundaries in place when custody is one day a week and she has a mother who will not support those boundaries in any way is a near impossible task.
DSD needs to understand the potential implications of her actions, even at 13 years, and I think her father should sit her down and explain exactly why she should never do anything like this again. I think there should be consequences too- grounding or no privileges for a period of time for example. And an apology to her step-sister- I always found having to acknowledge the wrong-doing worse than any other punishment.
I also don't agree the OPs DD's happiness and sense of security in her home should be sacrificed for the sake of the DSD. Both girls needs are important, and so many posters don't even seem to think OPs DD needs matter at all. I find this odd.
In my view, I would not "ban" DSD from the home but I would make it very clear what is and is not acceptable and what punishment will be in place for bad behaviour and ask that she apologises to your DD for what she has done. I would be keeping a close eye, and if it is negatively effecting your DD, you may need to take further steps to protect her, such as ensuring little contact between them. I think you should also suggest your DH tries to discuss things with this ex-W to come up with mutually agreed strategy if this is possible. If this doesn't work/isn't possible, I think he may need to seek advice from professionals about what to do next