Solution:So, after a significant amount of time trying to alter the original Process Annotations shape, I eventually gave up on that and decided to use a workaround to create an original shape instead. At first, I wanted to avoid doing this, because it's a little clunky, but the results made it worth it in the end.
I started by creating a basic box shape, then making it a group. With that in place, the group shape would hold the "cc" addon and create the association with the shape that I wanted to display the data from. Now, when the addon runs, it creates text fields and populates them. This was the problem that I was dealing with because I was struggling to "modify" those fields to have them tabbed and formatted as I wanted.
So I just used Hide Text on the group shape and locked away its text edit options.
Then I was able to pull the target shape's name and index the "ccReportOn" list to create cell references within the shape data section. I used those shape data cells in the group shape's shape sheet to create my own text fields in the sub shape. Since the "cc" addon wasn't interfering with the text in the sub shape, I formatted everything as needed and added the horizontal and vertical lines.
The end result is a call-out shape that displays the associated shapes data using the Configure Callout addon that adjusts its label and description columns based on the longest text string in each. I also set it up to auto-align with the associated shape. It looks simple, but its usability in my application is priceless.

I never would have imagined that this would be quite as complicated as it was. All and all happy with it, but I wish I wouldn't have had to resort to the hidden text option. It bothers me, but my end users will never know the difference unless they get into shape sheet. I'm hesitantly going to share it here in case anyone else would like to try and use it, but it's likely far from perfect and it's very specific to my application. But someone may find it useful and I'm open to constructive criticisms of course.
EDIT: The attached shape seems to be functioning incorrectly when added to a page with scaling. I'm looking into the issue, but I'm sure it has something to do with the resizing based on text. This is a new one for me, so there are some kinks to work out.
EDIT2: I corrected the issue. Seems I had some bad math happening in the tabs section. The formula was converting the pt measurement to inches, so AntiScaling caused some poor functionality. I forced the formula to use the original pt measurement and removed antiScale from the tabs. This seems to have fixed the issue... Until something else comes up I'm sure. The updated copy is attached.
If someone has a better way to handle this tabbing functionality, I'd love to get some feedback.