You have my empathy as it is a difficult situation for all concerned. Like others have said sometimes the best option is to keep your own counsel.
However, in your post you say your GD "Feels that her mum, you DD, is going our of her way to Put Down your DGD."
So for your own peace of mind I would suggest that you step back and Listen - really Listen to what your DD is saying to your DGD.
In situations like this, especially with teenagers, "Feelings" run high. If you stop and "Listen" you will be better placed to "Think" calmly if your DD really is "Putting your GD Down".
If there really is "Put Downs" coming from your DD then IF, and I do stress IF, you feel you want to address the way your DD is putting across her point to your GD you need to consider how you will approach that conversation with your DD.
I would suggest you ask yourself some questions about what your DD is saying to your GD.
Questions such as :- Does your DD have a real and valid point to make within what she is saying to your GD?
Is your GD correct that her Mum is just "Putting her Down" ?
If your DD has got good reason for what she is saying to your GD is she getting her point across in the best way?
Can you suggest a better way of making the point so your DD and GD can find a better way of communicating?
Is your GD's behaviour in need of guidance/adjustment from her Mum?
If so can you back your DD up by breaking it down and explaining things to your GD?
If your GD is correct and your DD is "just" nagging/putting your GD down you might want to "Listen" to "What" is being said - what is the real message within the words your DD is using - than ask yourself what is behind those words.
Ask yourself -with your knowledge of your DD life - what stress, dis-stress, fear, care, anger, hurt might be behind your DD approach to your GD.
Then "Think" very, very carefully what you do with what you have learned. Not only about the "What" you do but think very carefully about "How" you approach both your GD and your DD - IF you do anything at all.
What I encourage you to do is to get "Feelings" out of the situation for a while and let "Thinking" and "Listening" in. You can bring calm, balance, reason to the relationship between your DD and your GD.
Once you have a clearer understanding of "Where" each of them have a valid point and you have a better insight into "How" those points are getting lost in the "Emotions" of the situation you may feel more balanced yourself and once you are clearer you will have a lot more to offer.
It's not that "Feelings" aren't valid because they are. And teenagers, Bless 'Em, have lots and lots of "Feelings" and haven't learned to balance those 'Feelings" with "Thinking".
Sometimes it's wise to step back from "Feelings" and put some "Thinking" into the situation in order to sort out what's important and then hurt "Feelings" can be appropriately addressed.
I wish you a successful outcome for all of you in this difficult situation and I hope you find some peace for yourself in your concerns.