Solution
TestStand Deployment Utility handles adding JSON toolkit differently with the source distribution. When it says name in the error message, it means the full qualified name (FQN) of certain VIs. You can have two VIs called MyVI.vi, but LabVIEW will understand they are different if they are in different libraries.
LibA.lvlib > MyVI.vi is different from LibB.lvlib > MyVI.vi
The same happens for VIs with class members with the same name but within different LabVIEW Classes
ClassA.lvclass > MyVI.vi is different from ClassB.lvclass > MyVI.vi
Nevertheless, you can also use Source Distribution, as long as you set the option to include the JSON files in a new LabVIEW Library.
Define a JSON Lib folder to accommodate all the TestVI.vi dependencies (in this case JSON Lib but it has other libs too). In addition to that, create a new LabVIEW project library and add all files dropped in the JSON Lib folder to it. This will allow LabVIEW to differentiate files.

The best practice recommends pulling all dependencies of your code module into the project and then properly separating them and grouping them in different libraries in the build specification.
By properly configuring the source distribution build specification, you should be able to build the deployment image, installer, and other components successfully.
