I don't think a mooncup would be noticeable, no, so that's a good point. Tampon strong would, like PPs mentioned, be noticed during catheterisation but I could see a mooncup going undetected for a while unless you had a full body scan that might show it. It would eventually overflow, but who would know it wasn't just your period starting?
I'm a nurse and although I don't work in hospitals, I'd be surprised if internal vaginal exams were part of routine ITU care.