Upgrade in-place with low disk space on C: drive
Can you do it? Yes, you can. Should you do it? No, no shouldn't.

You're playing with fire, less than 10% free space on C drive is an urgent/critical flag in any decent company.

If you spot anything worth deleting, go for it. But in any case, since this is a physical server and you said Wintel team can't help you, present the scenario as an issue and a possible emergency in the future.