I've never understood how you would empty a mooncup in work toilets/public toilets and not end up with blood all over your fingers and clothes, door handle, taps etc.
Sorry for TMI. How is it done?
As others said you may not actually need to empty while out, but in case you do, you could look for a larger toilet that has a sink inside, most larger toilets have one (not disabled loo) or if you know of somewhere that only has one public toilet that functions as men/women/disabled. Then you have access to a private sink.
Otherwise wipe fingers with toilet paper so that blood does not get on your clothing, if there is some visible dried blood on fingers (usually around nails), hide them in sleeve or tuck fingers into a fist while you walk to the sink. Once at the sink with water flowing nobody is looking that closely at your hands and can't see anyway. As it's a fresh stain, it washes off easily.
Sadly, my mooncup doesn’t fit the same after I gave birth so I really miss it.
You can get different sized cups, look at different brands. You might need a larger or smaller cup, or a firmer or softer one.