Win.
With the cup, either a damp paper towel or a little water spray bottle instead of a public sink. The latter in countries that don’t have potable tap water.
Rinsing after dumping the cup during the day is actually optional. It should be cleaned once a day (I use soap and water, 'cause it doesn’t irritate me, but you can buy specially hooha pH balanced cleansing solution if you like) but it doesn’t HAVE to be cleaned every time you empty it if you need to empty it multiple times a day.
That being said, water bottles with a sports top rock.
Is any tampon made with a synthetic string, like nylon filament, instead of absorbent cotton? Would that solve the problem? It would be a nuisance, but you could buy some fishing line, and cut a bunch of 6 inch pieces, and tie one to the base end of each tampon string, and then cut off the cotton string provided.
Its been 25 years since I had to worry about any of that. Gawd I am so glad for age!