Derek,
Thank you for confirming that this is not currently possible. I upvoted the Idea request.
I did, thankfully, realize a workaround for now. Since it's based on the first ObjectID, I just added a blank row and then cut/paste the original data back into the feature class. This did the trick and now the Details window is blank once I open the Dashboard.
This is a Public Dashboard and you're welcome to take a look at it here:
Mississippi Forestry Commission Community Awareness Dashboard