Depending on the bathroom layout... I would not think twice about emptying my cup in the toilet then rinsing in the sink. It's what I do at home, and I would be kind of curious if someone used the toilet in our house but I then heard the shower running.

That said, I know my bathroom back home in Australia has the toilet separated from the bathroom and its sink, as do other friends' houses here - in that case, I'd empty the cup in the toilet and give it a bit of a wipe down with toilet paper, then wash it with hot water next chance I got. (In the shower if I'm home.)