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That's not really a good example as that too will rely on the capabilities which we already know seem to be blocked. Going to the Profile > User Groups may help a little more to see what the user is being set as...

So as expected, SP knows the user is an SP Admin on the front end where there is no influence being exerted.

Unless anyone else has any other ideas I really think this is an issue you need to discuss with the authors of the other plugin. My best guess and assumption is that for some reason it is blocking the WP capabilities that are set up for that user. Why it does not block yours as well is another good question. Can you ask them? We would be most interested in what they say in response.