If I grab the little target icon and drag it onto an Edge window/tab in the UI it always shows the same process id (the main MicrosoftEdge.exe process) and nothing is ever captured. Instead I have to manually go to the filtering tab, use the "Show only traffic from" drop down and figure out which MicrosoftEdgeCP.exe instance is the tab I'm interested in and filter it there.
I'm not sure what all you'll be able to do about this short of detecting it's MicrosoftEdge.exe and doing something more specific, but I thought I'd raise the issue to make sure you're aware of it.