Breast milk ice cubes?

My daughter is teething really badly! She is a hot mess. She really likes putting cold things on her gums. I wanted to put breast milk ice cubes in her silicone feeders to help her. We have been using frozen washcloths but they thaw so quickly! I can’t keep up!

Question: the only expressed milk I have is frozen. I can’t thaw it, make the cubes, and then refreeze it can I? I was pretty sure you can’t refreeze milk once it is thawed. I can always keep my frozen milk to mix with purées. Should I just pump some fresh milk and make the cubes?

My only concern, is I don’t usually pump, and my daughter is teething, which is already messing with my supply. I don’t want to screw it up lore by pumping. 🤦‍♀️