I have another use that's a lot of fun! If you have a dehydrator, you can make flax seed crackers and put the currants in. There's a company that makes something called "flackers" (flax seed crackers). This morning I tried the cinnamon currant flavor and it was delicious. Paleo friendly and even better if you make them yourself!
I'd freeze them for when the season's over, or roast them down to create a nice thick compote that you can add to things as needed!
OMG, they were one of the original fruits used to make European Rote Grütze! I would cook up a big batch using the currents in place of the cherries in my Paleo Rote Grütze recipe here.
My advice is to freeze them and then use them in smoothies. There are lots of black currant smoothie recipes on google. However, you'll have to make substitutions to keep it paleo. Most call for yogurt; I use spinach and/or almond milk instead. Also, if you have a juicer with a sorbet plate/screen, you can also use the frozen ones to make a paleo ice cream. You may be able to get away with just the black currants or you may end up adding coconut milk, depending on your juicer's recommended instructions.
Best of luck! Enjoy!
Just eat them. Your ancestors would eat them every day for a month in season, and be none the worse for it.