You don't care about his MCSE status. You want to get rid of him. Stop that attitude right now. The examples you cite aren't a very convincing reason to fire a temp.
Instead, give him tasks to accomplish. Note the amount of time you spend helping him. Note whether you need to help him with the same issues over and over again. Note the mistakes he made that you needed to intervene and fix.
With this information, you can build a metric that proves whether or not he's effective in his current role -- helping the company with IT backlog. All you need to do is show that it costs more of your time to keep him long term than to get rid of him.
There's no need to try to back him into a corner about his skills. He was hired, the credentials test is over. What matters is whether or not he can deliver. You don't start grilling him with MCSE test questions.
And let's say you did prove what you need to . . . What's next? You go to management with them? I think this is a huge mistake and you should abandon all plans to catch this guy as an impostor.
Your posts have a few red flags. I'll list the biggest one last:
It leads me to believe you're not the easiest person to work with right now.
Most organizations that have an unhealthy view of IT wouldn't go out of their way to hire a new member of the department. So a company would listen to you when you complain about his competency. They definitely would have had you sit in on the interview. So even if my assessment is wrong and you're a fully happy member of the team, there seems to be a perception that something's up with you. If that's true, you shouldn't come across as nit-picky but practical. Practical is citing how bad he is at his job even after a bit of training.
Furthermore, most people who have a dysfunctional team member complain about the many things they do wrong and other annoyances. Like "I have to show them everything a million times!" Not claiming the person lied about credentials. It reads as petty to me. He didn't know how to map a network drive . . . Does he remember how your organization does it now?
I mean, think about it . . . Would someone feel comfortable or confident around you if you grill them or look down on them when they mention they don't know something you think "someone of their status" should know? Especially if follow up questions outright demonstrate you don't believe he's honest. Especially (for your sake) if the temp could reasonably show that the questions were in the context of your organization.
"I know how to map drives a few different ways in Group Policy. I wanted to make sure I was using a consistent method." Is it true? I dunno. Is it true to management who hired this guy without consulting you and doesn't care about your opinion? Probably.
But that's neither here nor there. The third degree would make me feel like really uncomfortable. I'd confront you about it. Some people would go to HR. Some people would back away from the situation and avoid you when they perceive you're being angsty towards them. He's not a student to be tested for your approval. He already went through the approval process. And passed. Backtracking now seems disrespectful not only to him but to the people that hired him. You're essentially saying the person who hired the temp did their job wrong.
That's why I'm advocating you not do that. You accept this temp as an olive branch rather than a thorny bramble. You give him projects that should free up your time. You track your time not his time. And after a few weeks or so, raise a concern that fewer projects are getting completed lately because you always feel like you're helping the temp. But you don't present this as an attack, but as a business concern. The projects he's doing won't be as important as the projects you're focused on. And bring it back to dollars and cents . . . The tasks that save the company the most money are being put off for the ones a temp can handle. That would be the brunt of the business concern.