Hi Kyle,
I would recommend to write any data you want to persist across projects to this folder:
var settingsPath = System.IO.Path.Combine (Environment.GetFolderPath(System.Environment.SpecialFolder.CommonDocuments),
"ArcGIS", "ArcGISPro", "Settings");
ArcGIS Pro is already using the "ArcGIS\ArcGISPro" folder in the common documents folder across projects. You can read/write your settings from/to a file in that common location. I am not sure about you html file question. I don't think you can read a json file from within an html file that is stored on your local disk, I am pretty sure this results in some type of access exception. However, you could write your configuration in form of html so that it is human readable and parse the file in order to get your configuration settings.