I have experienced this somewhat from this side of your DIL. I know you mean well and I do understand your frustration, but you are taking the wrong approach here and it will NEVER work. Trust me on this.
I am no contact with my twin sister because she has been very hurtful to me. She has not apologised to me, and whilst my parents accept that she has behaved very badly they continue to minimise her behaviour. They desperately want us to reconcile, probably to make them feel better, but can't accept what I need for that to happen and want me to just 'move on'. Note that I have said NEED, not want.
Your DD has severely hurt your DIL. Intention has nothing to do with it. You, your DD or anyone else have absolutely no right to minimise or invalidate her feelings. Your DIL is absolutely within her rights to not speak to your DD and it is not your place to tell her otherwise. By standing by your DD regardless of right and wrong you are showing your DIL and your DS that you don't value their feelings.
The only way forward is your you DD to take the time to understand what she has done and sincerely apologise. Your latest update saying she hasn't had the chance to apologise is just an excuse probably because she is ashamed/stubborn/doesn't truly want to understand this from your DILs side. If she REALLY wanted to get in touch and apologise, she absolutely could. I know a true apology is hard, but that's for your DD to take on.
For example, have you spoken to your DIL and asked how she feels? When she's told you, have you listened and validated this, or have you given excuses or reasons, been defensive or role her she shouldn't feel that way?
For what it's worth, overhearing someone close to you mocking you behind your back is extremely painful and damaging. It's not just what is said, it's the broken trust and invalidation of how you previously viewed that relationship, plus the anxiety around what else has or will be said. It doesn't matter if your DD or don't agree with this. Other people experience things differently and there is no set standard or manual. She is not 'too sensitive'.
I recommend you and your DD look up 'Jimmy On Relationships' or similar accounts on Instagram or Facebook. He helps to show what a real apology is, and why validating someone's feelings and experiences is important, and how relationships can't survive if without this. He focuses on romantic ones but the concept is the same. There are many many other accounts too. You could also Google validation and proper apologies.
I know you mean well and want everyone to play happy families but the only way for this to happen is for you to honestly and proactively see this from your DILs side and give her what SHE NEEDS to move on. If she still can't after a proper apology and discussion over what level of relationship she may be able to start with, then you need to respect that. Then you can address seeing her separately if you wish.
You may want to send your DIL a message apologising for not listening to her pain, and tell her that you are now here for her. Ask her honestly how she feels, listen, and sympathise. Don't answer back. Ask her if there is a way forward, and how you can support her in this. Once your DD is also in a place to apologise and listen openly and honestly to your DIL, your role could be to help mediate, but only from the view of ensuring your DIL is able to heal.
https://www.instagram.com/jimmy_on_relationships?igsh=N2Fheno0OWpwZm52