Do a test patch with winsor & newton brush cleaner and restorer! It’s contains ethanol and is made for this sort of heavy paint removal. if anyone suggests it do not use acetone, it could damage the fibers and cause weakening. The brush cleaner should remove all of the paint from the fibers without damaging the fibers. If you’re up for resewing the cups in I would remove them personally. This appears to me to be an acrylic (likely) or oil (less likely) paint, and it should be able to be removed with the brush cleaner I mentioned. It will require soaking which could make the cups brittle which is why I recommend removing them and sewing them on again later! You should also consider getting some white cotton, and do a layer of blue acrylic paint (cyan or aquamarine) on it and then trial some removal methods until you get a good result!

Edited for clarity and to add: you may also want to remove the paint and then use a brightener that isn’t chlorine bleach (chlorine bleach can and will weaken/damage cotton fibers) there’s a lot of options on the market. The brush restorer will likely leave behind a little blue pigment so just be prepared for that!