How to clean menstrual cup in public bathrooms?

Since having my baby, I saw a gynaecologist because I couldn’t put tampons in anymore, they would just fall out straight away. The gynaecologist said that nothing was wrong inside of me, no prolapse and stuff. My daughter is nearly 2 years and I still can’t put a tampon up. I despise pads with a passion, they irritate me loads even organic ones. So I got a menstrual cup and for some reason that fits up there and stays. Sometimes it does come down lower than I put it in but it never falls out like the tampons. Now I’m starting to use them more, I was wondering how to wash them in a public bathroom when out? Or if anyone does that? Because I normally will take my cup out before leaving home and switch to a pad so it’s easy to change. It’s still irritating me but I don’t think I would want to wash the cup in a sink with others around. I’ve never walked into a public bathroom and saw someone doing that so I assumed women don’t. I also have a steriliser for them and a second cup that you can sterilise and put in the case ready to go. Does anyone else use the cups and clean them in the sink? It might sound stupid but not every woman would want to see the blood in the sink I don’t think and I don’t want to go and do it and offend people or something.