I'm born & raised in South Florida and believe it or not, sometimes the salt water and sun actually HELP miscellanous rashes and whatnot. I don't know the science behind it (salt water is like an antiseptic and can help dry things out, i know) and the sun as well. When my daughter was little she got this random rash somtimes and a weekend at the beach usually helped clear it up. Since your daughter has no fever or itching, this may be an option.
You also might want to take away her sippy cup when she plays with it like that. It might be having the same effect as thumb sucking or a pacifier would in keeping the area on her chin too moist, therein causing the bumps.
Just some thoughts ... Best of luck to you!!!
C.